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"798": {
"sentence.id": 798,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "This 8-bit console had many different games in its day, so Rolfe had a massive library from which to draw.",
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"799": {
"sentence.id": 799,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "Whether you play platform games on rival 8-bit systems or thrive on the action of modern, expansive 3D games, the best Sega games will keep you hitting 'Yes' when the \"Continue?\" question appears.",
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"800": {
"sentence.id": 800,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "Approximately the size of the Game Boy Pocket, the Game Boy Color sported twice the processor speed and memory of its predecessor and allowed games to be made in glorious 8-bit color.",
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"sentence.id": 801,
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"sentence.sentence": "Both systems were 16-bit, representing a significant increase over the 8-bit era which preceded it.",
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"sentence.id": 802,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "After the 8-bit NES, we saw the 16-bit Sega Genesis, 32-bit Sony PlayStation, and 64-bit Nintendo 64.",
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"803": {
"sentence.id": 803,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "The backup processor was an 8-bit 4MHz ZiLOG Z-80A.The graphics were bright and vibrant, with a 65,000 color palette, allowing up to 4096 colors to be displayed on the screen at any given time.",
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"804": {
"sentence.id": 804,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "The Nintendo Entertainment System completely dominated the 8-bit era of home gaming.",
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"805": {
"sentence.id": 805,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "In this way, it initially competed head to head against the NES, and as such, had far superior graphics and sound compared to its 8-bit counterpart from Nintendo.",
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"806": {
"sentence.id": 806,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "From the Game Boy tin sound to the 8-bit sound of the Commodore 64, no version of Tetris was safe.",
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"807": {
"sentence.id": 807,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "If you go into your favorite search engine and look for a video game scene creator, there's a good chance that one of your beloved 8-bit system video games will have an online creation tool.",
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"808": {
"sentence.id": 808,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "In uncompressed 8-bit images, the 256 colors that create the image are referred to as a palette, index or is also known as a color lookup table (CLUT).",
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"809": {
"sentence.id": 809,
"sentence.ts": "2022-12-13 05:45:19",
"sentence.sentence": "While it's mostly obsolete today, there are still some uses for 8-bit color theory.",
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}
},
"659": {
"word.id": 659,
"word.ts": "2022-11-08 14:29:03",
"word.word": "8-foot",
"word.letter": "#",
"word.synonym_id": {},
"word.sentence_id": {
"819": {
"sentence.id": 819,
"sentence.ts": "2022-12-13 05:45:28",
"sentence.sentence": "He had meantime (in 1742) been appointed to succeed Edmund Halley as astronomer royal; his enhanced reputation enabled him to apply successfully for an instrumental outfit at a cost of 1000; and with an 8-foot quadrant completed for him in 1750 by John Bi",
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"820": {
"sentence.id": 820,
"sentence.ts": "2022-12-13 05:45:28",
"sentence.sentence": "Later, to clinch the doubles, he ran a good sweaty 8-foot rover.",
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"821": {
"sentence.id": 821,
"sentence.ts": "2022-12-13 05:45:28",
"sentence.sentence": "Using a table saw, cut the 8-foot piece of plywood into one long section that is 25 inches in width.",
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}
},
"705": {
"word.id": 705,
"word.ts": "2022-11-08 14:42:11",
"word.word": "x",
"word.letter": "x",
"word.synonym_id": {
"25": {
"synonym.id": 25,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "scratch",
"synonym.definition": "To strike out or cancel (a word, for example) by or as if by drawing lines through.",
"synonym.title": "",
"synonym.description": ""
},
"26": {
"synonym.id": 26,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "cross",
"synonym.definition": "Something that combines the qualities of two different things or types",
"synonym.title": "",
"synonym.description": ""
},
"27": {
"synonym.id": 27,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "tenner",
"synonym.definition": "(Brit.) A ten-pound note",
"synonym.title": "",
"synonym.description": ""
},
"28": {
"synonym.id": 28,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "unknown-quantity",
"synonym.definition": "A person or thing whose nature or value is a mystery.",
"synonym.title": "",
"synonym.description": ""
},
"29": {
"synonym.id": 29,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "strike",
"synonym.definition": "(Intransitive) To set off on a walk or trip.",
"synonym.title": "",
"synonym.description": ""
},
"30": {
"synonym.id": 30,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "cristal",
"synonym.definition": "street names for methylenedioxymethamphetamine",
"synonym.title": "",
"synonym.description": ""
},
"31": {
"synonym.id": 31,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "mystery",
"synonym.definition": "Something unexplained, unknown, or kept secret",
"synonym.title": "",
"synonym.description": ""
},
"32": {
"synonym.id": 32,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "y",
"synonym.definition": "Any of the speech sounds that this letter represents, as, in English, the semivowel (y) at the beginning of a syllable, as in yes or beyond, or the vowel (i) of myth, (ē) of holy, or (ī) of my",
"synonym.title": "",
"synonym.description": ""
},
"33": {
"synonym.id": 33,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "times",
"synonym.definition": "A person's experiences or biography.",
"synonym.title": "",
"synonym.description": ""
},
"34": {
"synonym.id": 34,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "Also used with out: annul",
"synonym.definition": "To remove or invalidate by or as if by running a line through or wiping clean.",
"synonym.title": "",
"synonym.description": ""
},
"35": {
"synonym.id": 35,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "x",
"synonym.definition": "A movie rating indicating that admission will not be granted to anyone under the age of 17.",
"synonym.title": "",
"synonym.description": ""
},
"36": {
"synonym.id": 36,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "decade",
"synonym.definition": "A group of ten",
"synonym.title": "",
"synonym.description": ""
},
"37": {
"synonym.id": 37,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "ten",
"synonym.definition": "Totaling one more than nine",
"synonym.title": "",
"synonym.description": ""
},
"38": {
"synonym.id": 38,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": 10,
"synonym.definition": "the cardinal number that is the sum of nine and one; the base of the decimal system",
"synonym.title": "",
"synonym.description": ""
},
"39": {
"synonym.id": 39,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "unknown",
"synonym.definition": "(Mathematics) A quantity of unknown numerical value.",
"synonym.title": "",
"synonym.description": ""
},
"40": {
"synonym.id": 40,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "variable",
"synonym.definition": "The definition of a variable is something that can change, or is a quantity in an equation that can change its value.",
"synonym.title": "",
"synonym.description": ""
},
"41": {
"synonym.id": 41,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "n",
"synonym.definition": "N is the fourteenth letter in the English alphabet.",
"synonym.title": "",
"synonym.description": ""
},
"42": {
"synonym.id": 42,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "ecstasy",
"synonym.definition": "A feeling of overpowering joy; great delight; rapture",
"synonym.title": "",
"synonym.description": ""
},
"43": {
"synonym.id": 43,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "xtc",
"synonym.definition": "street names for methylenedioxymethamphetamine",
"synonym.title": "",
"synonym.description": ""
},
"44": {
"synonym.id": 44,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "disco-biscuit",
"synonym.definition": "(Slang) A tablet of the drug ecstasy.",
"synonym.title": "",
"synonym.description": ""
},
"45": {
"synonym.id": 45,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "hug drug",
"synonym.definition": "street names for methylenedioxymethamphetamine",
"synonym.title": "",
"synonym.description": ""
},
"46": {
"synonym.id": 46,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "ex.",
"synonym.definition": "Ex is slang and is defined as a past partner.",
"synonym.title": "",
"synonym.description": ""
},
"47": {
"synonym.id": 47,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "letter",
"synonym.definition": "A written or printed communication, generally longer and more formal than a note.",
"synonym.title": "",
"synonym.description": ""
},
"48": {
"synonym.id": 48,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "mark",
"synonym.definition": "A number, letter, or symbol used to indicate various grades of academic achievement:",
"synonym.title": "",
"synonym.description": ""
},
"49": {
"synonym.id": 49,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "signature",
"synonym.definition": "(Figurative) A mark or sign of implication.",
"synonym.title": "",
"synonym.description": ""
},
"50": {
"synonym.id": 50,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "blot",
"synonym.definition": "To erase or get rid of",
"synonym.title": "",
"synonym.description": ""
},
"51": {
"synonym.id": 51,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "cancel",
"synonym.definition": "To cross out with lines or other marks, as in deleting written matter or marking a check as used and cleared",
"synonym.title": "",
"synonym.description": ""
},
"52": {
"synonym.id": 52,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "delete",
"synonym.definition": "To take out (a printed or written letter, word, etc.); cross out",
"synonym.title": "",
"synonym.description": ""
},
"53": {
"synonym.id": 53,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "efface",
"synonym.definition": "To erase (as anything impressed or inscribed upon a surface); to render illegible or indiscernible.",
"synonym.title": "",
"synonym.description": ""
},
"54": {
"synonym.id": 54,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "erase",
"synonym.definition": "To remove (something written, for example) by rubbing, wiping, or scraping.",
"synonym.title": "",
"synonym.description": ""
},
"55": {
"synonym.id": 55,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "expunge",
"synonym.definition": "To erase, delete, or strike out:",
"synonym.title": "",
"synonym.description": ""
},
"56": {
"synonym.id": 56,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "obliterate",
"synonym.definition": "To render invisible or unreadable, as by erasing or marking over:",
"synonym.title": "",
"synonym.description": ""
},
"57": {
"synonym.id": 57,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "rub",
"synonym.definition": "To move (one's hand, a cloth, etc.) over, or spread or apply (polish, etc.) on or over, a surface or object in this way",
"synonym.title": "",
"synonym.description": ""
},
"58": {
"synonym.id": 58,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "undo",
"synonym.definition": "To reverse the doing of (something accomplished); do away with; cancel; annul",
"synonym.title": "",
"synonym.description": ""
},
"59": {
"synonym.id": 59,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "wipe",
"synonym.definition": "To rub or pass over with a cloth, mop, etc., as for cleaning or drying",
"synonym.title": "",
"synonym.description": ""
},
"60": {
"synonym.id": 60,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "vacate",
"synonym.definition": "To leave an office or position.",
"synonym.title": "",
"synonym.description": ""
},
"61": {
"synonym.id": 61,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "adam",
"synonym.definition": "(Name, person, proper) A masculine name",
"synonym.title": "",
"synonym.description": ""
},
"62": {
"synonym.id": 62,
"synonym.ts": "2022-12-04 07:54:45",
"synonym.synonym": "go",
"synonym.definition": "(Brit., Informal) A quantity given or taken at one time",
"synonym.title": "",
"synonym.description": ""
}
},
"word.sentence_id": {
"829": {
"sentence.id": 829,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The swellings have been found to be due to a curious hypertrophy of the tissue of the part, the cells being filled with an immense number of minute bacterium-like organisms of V, X or Y shape.",
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"830": {
"sentence.id": 830,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In general his object is to reduce the final equation to a simple one by making such an assumption for the side of the square or cube to which the expression in x is to be equal as will make the necessary number of coefficients vanish.",
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"831": {
"sentence.id": 831,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "See The Imperial Gazetteer of India (Oxford, 1908), x.",
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"832": {
"sentence.id": 832,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus, i part by weight of hydrogen unites with 8 parts by weight of oxygen, forming water, and with 16 or 8 X 2 parts of oxygen, forming hydrogen peroxide.",
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"833": {
"sentence.id": 833,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Again, in nitrous oxide we have a compound of 8 parts by weight of oxygen and 14 of nitrogen; in nitric oxide a compound of 16 or 8 X 2 parts of oxygen and 1 4 of nitrogen; in nitrous anhydride a compound of 24 or 8 X 3 parts of oxygen and 14 of nitrogen;",
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"834": {
"sentence.id": 834,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Di-derivatives x x x p v as $ v as s Here we have assumed the substituent groups to be alike; when they are unlike, a greater number of isomers is possible.",
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"835": {
"sentence.id": 835,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thomsen deduces the actual values of X, Y, Z to be 14.71, 13.27 and zero; the last value he considers to be in agreement with the labile equilibrium of acetylenic compounds.",
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"836": {
"sentence.id": 836,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It will be seen that (1) the increase in equivalent volume is about 6.6; (2) all the topic parameters are increased; (3) the greatest increase is effected in the parameters x and tG, which are equally lengthened.",
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"837": {
"sentence.id": 837,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The words ascribed to Christ in Luke x.",
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"838": {
"sentence.id": 838,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Such curves are given by the equation x 2 - y 2 = ax 4 -1bx 2 y 2 +cy 4 .",
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"839": {
"sentence.id": 839,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the very early rock inscriptions of Thera (700-600 B.C.), written from right to left, it appears in a form resembling the ordinary Greek X; this form apparently arose from writing the Semitic symbol upside down.",
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"840": {
"sentence.id": 840,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus if a person holds futures for 10,000 bales which stood at 5.20 on the last settlement day and now stand at 5.30, and in the course of the previous week has sold 5000 bales of \" futures \" at 5.1 o, he receives 10,000 X - i ce o d.",
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"841": {
"sentence.id": 841,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "During 1822 and the succeeding years he travelled about Europe on the search for materials for his Collection des chroniques rationales fran4aises ecrites en langue vulgaire cat XIII e au X VI' siècle (47 vols., 1824-1829).",
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"842": {
"sentence.id": 842,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In any case, since Ben-Sira belongs to about 180 B.C., the date of' Koheleth, so far as these coincidences indicate it, would not be far from 200 B.C. The contrast made in x.",
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},
"843": {
"sentence.id": 843,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The \" Long Walls \" (Ta µaKpet TEixn, Ta cnc X) consisted of (1) the \" North Wall \" (TO l36p€tov TEIXor), (a) the Walls.\"",
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"844": {
"sentence.id": 844,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The arch is surmounted by a triple attic with Corinthian columns; the frieze above the keystone bears, on the north-western side, the inscription aZS' 'Aqvat, OouEw 7rpiv rats, and on the south-eastern, aZS' do' `ASptavoii Kai ou X i Ono-Los 'TO Xis.",
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"845": {
"sentence.id": 845,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' ",
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"846": {
"sentence.id": 846,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A, Five specimens of Echinorhynchus acus, Rud., attached to a piece of intestinal wall, X 4.",
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"847": {
"sentence.id": 847,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Some, however, see in it a corruption of the Semitic name samekh, the letter which corresponds in alphabetic position and in shape to the Greek (x).",
"sentence.uniq": "20777661bd164d287836ec5575c913f0"
},
"848": {
"sentence.id": 848,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Coote's Remarkable Maps of the X Vth, X Vlth and X VIIth Centuries reproduced in their Original Size (Amsterdam, 1894-1897), and Bibliotheca lindesiana (London, 1898) with facsimiles of the Harleian and other Dieppese maps of the 16th century.",
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"849": {
"sentence.id": 849,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Also a unit class is any class with the property that it possesses a member x such that, if y is any member of the class, then x and y are identical.",
"sentence.uniq": "b5adb3631b08162eae992923011507de"
},
"850": {
"sentence.id": 850,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A doublet is any class which possesses a member x such that the modified class formed by all the other members except x is a unit class.",
"sentence.uniq": "035343ebce1cbf71a03d4383110bf2de"
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"851": {
"sentence.id": 851,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A relation (R) is serial when (I) it implies diversity, so that, if x has the relation R to y, x is diverse from y; (2) it is transitive, so that if x has the relation R to y, and y to z, then x has the relation R to z; (3) it has the property of connexit",
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"852": {
"sentence.id": 852,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Two relations R and R' are said to be ordinally similar, if a one-one relation holds between the members of the two fields of R and R', such that if x and y are any two members of the field of R, such that x has the relation R to y, and if x' and y are th",
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"853": {
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"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "R and R', x and x', &c. It is evident that the ordinal similarity of two relations implies the cardinal similarity of their fields, but not conversely.",
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"854": {
"sentence.id": 854,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.",
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"855": {
"sentence.id": 855,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If a is a real number, +a is defined to be the relation which any real number of the form x+a bears to the real number x, and - a is the relation which any real number x bears to the real number x+a.",
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"856": {
"sentence.id": 856,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If such a complex number is written (as usual) in the form x i e l +x 2 e 2 +...",
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"857": {
"sentence.id": 857,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The product of two complex numbers of the second order - namely, l e l +x 2 e 2 and y i e l +y 2 e 2, is in this case defined to mean the complex (x i y i - x 2 y 2)e i +(x i y 2 +x 2 y 1)e 2.",
"sentence.uniq": "7d8e6298836844045c78e966471ecbdb"
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"858": {
"sentence.id": 858,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Call this class w; then to say that x is a w is equivalent to saying that x is not an x.",
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"859": {
"sentence.id": 859,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Now consider a propositional function Fx in which the variable argument x is itself a propositional function.",
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"860": {
"sentence.id": 860,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Accordingly, it is a fallacy for any determination of x to consider \"x is an x\" or \"x is not an x\" as having the meaning of propositions.",
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"861": {
"sentence.id": 861,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Note that for any determination of x, \"x is an x\" and \"x is not an x,\" are neither of them fallacies but are both meaningless, according to this theory.",
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"862": {
"sentence.id": 862,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "His orders were at once issued and complied with with such celerity that by the 31st he stood prepared to advance with the corps of Soult, Ney, Davout and Augereau, the Guard and the reserve cavalry (80,000 men on a front of 60 m.) from Myszienec through ",
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"863": {
"sentence.id": 863,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The Flos of Leonardo turns on the second question set by John of Palermo, which required the solution of the cubic equation x 3 -{-2x'-}-lox = 20.",
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"864": {
"sentence.id": 864,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Perfectly pure distilled sea-water dissociates, to an infinitesimal degree, into hydrogen (H) and hydroxyl (HO) ions, so that one litre of such water contains 1 X 10 7, or 1 part of a gram-molecule of either hydr010,000,000 gen or hydroxyl (a gramme-molec",
"sentence.uniq": "ff83ecb3352851469b77d97bf501e882"
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"865": {
"sentence.id": 865,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Taking the chemical equivalent weight of silver, as determined by chemical experiments, to be 107.92, the result described gives as the electrochemical equivalent of an ion of unit chemical equivalent the value 1 036 X 5.",
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"866": {
"sentence.id": 866,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If, as is now usual, we take the equivalent weight of oxygen as our standard and call it 16, the equivalent weight of hydrogen is I o08, and its electrochemical equivalent is I 044 X 5.",
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"867": {
"sentence.id": 867,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let x be the number of molecules which dissociate per second when the number of undissociated molecules in unit volume is unity, then in a dilute solution where the molecules do not interfere with each other, xp is the number when the concentration is p. ",
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"868": {
"sentence.id": 868,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The number of undissociated molecules is then I - a, so that if V be the volume of the solution containing I gramme-molecule of the dissolved substance, we get q= and p= (I - a)/V, hence x(I - a) V =yd/V2, and constant = k.",
"sentence.uniq": "66c6faac370ef60fb6b27043f0a0e29d"
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"869": {
"sentence.id": 869,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The dynamical equivalent of the calorie is 4.18 X Io 7 ergs or C.G.S.",
"sentence.uniq": "7d40b47370d2db38b7ea9dd9346716dd"
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"870": {
"sentence.id": 870,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we take as an example a concentration cell in which silver plates are placed in solutions of silver nitrate, one of which is ten times as strong as the other, this equation gives E = o 060 X Io 8 C.G.S.",
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"871": {
"sentence.id": 871,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions 'X=' by+ cz, Y = a'x + b'y + c'z, Z =a\"x+b\"y-l-c\"z, where X 2+Y2+Z2 = x2+ y2+z2.",
"sentence.uniq": "93eb02c2b00bbe05f7d083848eba6abb"
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"872": {
"sentence.id": 872,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.",
"sentence.uniq": "3c6b0b65da185220e33b6e7b288898b7"
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"873": {
"sentence.id": 873,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From the differential coefficients of the y's with regard to the x's we form the functional.",
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"874": {
"sentence.id": 874,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Hence if A does not vanish x 1 = x 2 =...",
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"875": {
"sentence.id": 875,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear equations satisfied by the m+n quantities B1, 2, ...B n, A 1",
"sentence.uniq": "728ab0ffafc95b991d9b420ed5d0a8bc"
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"876": {
"sentence.id": 876,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being ",
"sentence.uniq": "ec1746f665f893be25f7018defa832e5"
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"877": {
"sentence.id": 877,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For such functions remain unaltered when each root receives the same infinitesimal increment h; but writing x-h for x causes ao, a1, a 2 a3,...",
"sentence.uniq": "15171cba362c539effdaaa8055cea4ef"
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"878": {
"sentence.id": 878,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Further, let 1 -1-b i x+ b 2 x 2' +...",
"sentence.uniq": "7d2528479b2b7764627ae82eee07d5c2"
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"879": {
"sentence.id": 879,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If m be infinite and 1 + b i x + b 2 x 2 +...",
"sentence.uniq": "2a1fbbb8ff6519099eec5138aac026a7"
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"880": {
"sentence.id": 880,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In this notation the fundamental relation is written (l + a i x +01Y) (I + a 2x+l32Y) (1 + a3x+133y)...",
"sentence.uniq": "58ddbd4937f204266b5ccea3cb9d65c4"
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"881": {
"sentence.id": 881,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It may be written in the form n n-1 2 ax 1 +bx1 x2 +cx 1 x 2 + ...; or in the form n n n=1 n n-2 2 +(1)bx x2+ ?",
"sentence.uniq": "261b5fcb03b7c17b81cdaa95948cab5f"
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"882": {
"sentence.id": 882,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "By solving the equations of transformation we obtain rE1 = a22x1 - a12x1, r = - a21x1 + allx2, aua12 where r = I = anon-anon; a21 a22 r is termed the determinant of substitution or modulus of transformation; we assure x 1, x 2 to be independents, so that ",
"sentence.uniq": "867ea86235a406e0a035b94d06df34e5"
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"883": {
"sentence.id": 883,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...",
"sentence.uniq": "54a4c52f078ab882d55f81b2816eadc3"
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"884": {
"sentence.id": 884,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Such an expression as a l b 2 -a 2 b i, which is aa 2 ab 2 aa x 2 2 ax1' is usually written (ab) for brevity; in the same notation the determinant, whose rows are a l, a 2, a3; b2, b 2, b 3; c 1, c 2, c 3 respectively, is written (abc) and so on.",
"sentence.uniq": "99f426a4af288448c1a1f23d3b4f2836"
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"885": {
"sentence.id": 885,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For this reason the umbrae A1, A 2 are said to be contragredient to xi, x 2.",
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"886": {
"sentence.id": 886,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For this reason the umbrae -a 2, a l are said to be cogredient to 5 1 and x 2.",
"sentence.uniq": "97ed3ab3f645169cdba8bed9959f976b"
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"887": {
"sentence.id": 887,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In general it will be simultaneous covariant of the different forms n 1 rz 2 n3 a, b x, ?",
"sentence.uniq": "f31042a65c9506fff3a5d5082a7cd01d"
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"888": {
"sentence.id": 888,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From the three equations ax = alxl+ a2x2, b.= blxl+b2x2, cx = clxi+c2x2, we find by eliminating x, and x 2 the relation a x (bc)+b x (ca) +c x (ab) =0.",
"sentence.uniq": "15e465fbe95e18e648aa91f53297795b"
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"889": {
"sentence.id": 889,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "X1, X 2, u1, /22 being as usual the coefficients of substitution, let x1a ?",
"sentence.uniq": "e77ebad30a8396490b0fe1a718e9c1af"
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"890": {
"sentence.id": 890,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2, -x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on.",
"sentence.uniq": "4b80375427d98ced92b6fff6d2e3e508"
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"891": {
"sentence.id": 891,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.",
"sentence.uniq": "fa3f730ab707fe853d7e2f448ccbba17"
},
"892": {
"sentence.id": 892,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From (ac) 2 (bd) 2 (ad)(bc) we obtain (bd) 2 (bc) cyd x +(ac) 2 (ad) c xdx - (bd) 2 (ad)axb x - (ac)2(bc)axbx =4(bd) 2 (bc)c 2.",
"sentence.uniq": "545d949364a6e19e4e563c4a18a9d9de"
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"893": {
"sentence.id": 893,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Similarly regarding 1 x 2 as additional parameters, we see that every covariant is expressible as a rational function of n fixed covariants.",
"sentence.uniq": "23b9ce322024e71339a424bd457929d0"
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"894": {
"sentence.id": 894,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "To exhibit any covariant as a function of uo, ul, a n = (aiy1+a2y2) n and transform it by the substitution fi y 1+f2 y where f l = aay 1, f2 = a2ay -1, x y - x y = X x thence f .",
"sentence.uniq": "4c8c029bd1b26ab54638f12ea6261661"
},
"895": {
"sentence.id": 895,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The first transvectant, (f,f') 1 = (ab) a x b x, vanishes identically.",
"sentence.uniq": "20115aaa1ed2adc958cb82bda053681c"
},
"896": {
"sentence.id": 896,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If (f,4) 1 be not a perfect square, and rx, s x be its linear factors, it is possible to express f and 4, in the canonical forms Xi(rx)2+X2(sx)2, 111(rx)2+1.2 (sx) 2 respectively.",
"sentence.uniq": "f2d36e1ee8cd8314b5c8fa4498e0565f"
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"897": {
"sentence.id": 897,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In fact, if f and 4, have these forms, it is easy to verify that (f, 4,)i= (A j z) (rs)r x s x .",
"sentence.uniq": "54af1b911ccd26a301c94425a28cd9d9"
},
"898": {
"sentence.id": 898,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The simplest form to which the quartic is in general reducible is +6mxix2+x2, involving one parameter m; then Ox = 2m (xi +x2) +2 (1-3m2) x2 ix2; i = 2 (t +3m2); j= '6m (1 - m) 2; t= (1 - 9m 2) (xi - x2) (x21 + x2) x i x 2.",
"sentence.uniq": "0ecd318df3fbf4309b5971937bb47f1e"
},
"899": {
"sentence.id": 899,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "T = (j, j) 2 jxjx; 0 = (iT)i x r x; four other linear covariants, viz.",
"sentence.uniq": "f7e901ed81a1d3709c6dfdaa189341cf"
},
"900": {
"sentence.id": 900,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When C vanishes j has the form j = pxg x, and (f,j) 3 = (ap) 2 (aq)ax = o.",
"sentence.uniq": "ad64ed627c4ebb37236def529ec7d06e"
},
"901": {
"sentence.id": 901,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For example, take the ternary quadratic (aixl+a2x2+a3x3) 2 =a2x, or in real form axi +bx2+cx3+2fx 2 x 3+ 2gx 3 x 1 +2hx i x 2.",
"sentence.uniq": "566b0f9c28e649ed7a2da972a4332e7b"
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"902": {
"sentence.id": 902,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.",
"sentence.uniq": "f64dea45cbdba60f662852eadda7a02f"
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"903": {
"sentence.id": 903,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The Hessian is symbolically (abc) 2 azbzcz = H 3, and for the canonical form (1 +2m 3)xyz-m 2 (x 3 +y 3 +z 3).",
"sentence.uniq": "90175c9bf0e8a351ed6a1eeabb33b21f"
},
"904": {
"sentence.id": 904,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the n",
"sentence.uniq": "f129db02f925715d81fe64a457ea3ca8"
},
"905": {
"sentence.id": 905,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely",
"sentence.uniq": "435890e5311d2df3883e95feb8b18009"
},
"906": {
"sentence.id": 906,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When R =0, and neither of the expressions AC - B 2, 2AB -3C vanishes, the covariant a x is a linear factor of f; but, when R =AC - B 2 = 2AB -3C =0, a x also vanishes, and then f is the product of the form jx and of the Hessian of jx.",
"sentence.uniq": "2f1c55969c09f887df11c5d83314229d"
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"907": {
"sentence.id": 907,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed a",
"sentence.uniq": "72dd9a791957079717d65253c37825b7"
},
"908": {
"sentence.id": 908,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Similarly, if 0 =3, every form (3K+12,x) is a perpetuant.",
"sentence.uniq": "e1ba0413ae64f875dd9b66b0e22eb660"
},
"909": {
"sentence.id": 909,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Forms.-Taking the two forms to be a o xi + pa l x i 1x2+p(p-1)a2xr2x2-I-...",
"sentence.uniq": "f1596cd2d12c47f0073afe7bfdb45ad7"
},
"910": {
"sentence.id": 910,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes.",
"sentence.uniq": "9e60e2787676f634145bad85742ccd29"
},
"911": {
"sentence.id": 911,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The substitution for x, y in terms of X, Y is the most general linear substitution in virtue of the four degrees of arbitrariness introduced, viz.",
"sentence.uniq": "0d29d2b9ef36d076a0a4d7e975676dc0"
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"912": {
"sentence.id": 912,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which",
"sentence.uniq": "e512c924c2f6646343a43c26b7ef5197"
},
"913": {
"sentence.id": 913,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.",
"sentence.uniq": "4471991673e928df70c6ae8446d5684c"
},
"914": {
"sentence.id": 914,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the a x = aixi+a2x2, observe that a a = a2, ab = aibi +a2b2.",
"sentence.uniq": "20bd1fcabfdfd0f2bf06e2fe73da62c5"
},
"915": {
"sentence.id": 915,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.",
"sentence.uniq": "f9abfd5681b7aaa574b98a6f533c3ea7"
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"916": {
"sentence.id": 916,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "To assist us in handling the symbolic products we have not only the identity (ab) cx + (bc) a x + (ca) bx =0, but also (ab) x x+ (b x) a + (ax) b x = 0, (ab)a+(bc)a s +(ca)a b = 0, and many others which may be derived from these in the manner which will b",
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"917": {
"sentence.id": 917,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of a x and x x is the line perpendicular to x and the vanishing of the quadrinvariant a x is the condition that a x passes through on",
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"918": {
"sentence.id": 918,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In general any pencil of lines, connected with the line a x by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero.",
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"919": {
"sentence.id": 919,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ax, or, a",
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"920": {
"sentence.id": 920,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If V denote the potential, F the resultant force, X, Y, Z, its components parallel to the co-ordinate axes and n the line along which the force is directed, then - sn = F, b?= X, - Sy = Y, -s Surfaces for which the potential is constant are called equipot",
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"921": {
"sentence.id": 921,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If F T is the force along r and F t that along t at right angles to r, F r =X cos 0+ Y sin 0=M 2 cos 0, F t = - X sin 0+ Y cos 0 = - r 3 sin 0.",
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"922": {
"sentence.id": 922,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Since 7ra'I is the moment of the sphere (=volume X magnetization), it appears from (10) that the magnetized sphere produces the same external effect as a very small magnet of equal moment placed at its centre and magnetized in the same direction; the resu",
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"923": {
"sentence.id": 923,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Hopkinson pointed out that the greatest dissipation of energy which can be caused by a to-and-fro reversal is approximately represented by Coercive force X maximum induction fir.",
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"924": {
"sentence.id": 924,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The actual experiment to which it relates was carried only as the point marked X, corresponding to a magnetizing force of 65, and an induction of nearly 17,000.",
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"925": {
"sentence.id": 925,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the pull is measured in pounds and the area in square inches, the formula may be written B =1317 X iI P/S +H.",
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"926": {
"sentence.id": 926,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If V is the volume of a ball, H the strength of the field at its centre, and re its apparent susceptibility, the force in the direction x is f= K'VH X dH/dx; and if K',, and are the apparent susceptibilities of the same ball in air and in liquid oxygen, K",
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"927": {
"sentence.id": 927,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "According to the notation adopted by Meyer the atomic susceptibility k=KX atomic-weight/ (density X 1000).",
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"928": {
"sentence.id": 928,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "About 2000 blue fox-skins were x.",
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"929": {
"sentence.id": 929,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The results, of which the most important are summarized in the article Sparta, are published in the British School Annual, x.",
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"930": {
"sentence.id": 930,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "His development of the equation x m +- px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.",
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"931": {
"sentence.id": 931,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We may regard this as meaning the same as 5 X3 X7 lb, since 7 lb itself means 7 X 1 lb, and the lb is the unit in each case.",
"sentence.uniq": "cc4aff12d36ec85d8656097edd81dcc1"
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"932": {
"sentence.id": 932,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "But it does not mean the same as 5 X 21 lb, though the two are equal, i.e.",
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"933": {
"sentence.id": 933,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This rule as to the meaning of X is important.",
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"934": {
"sentence.id": 934,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If it is intended that the first number is to be multiplied by the second, a special sign such as X should be used.",
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"935": {
"sentence.id": 935,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The sign X coming immediately before, or immediately after, a bracket may be omitted; e.g.",
"sentence.uniq": "c4c924f954aea21cc972b8e659c754e4"
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"936": {
"sentence.id": 936,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus 8+ (7 X 6) +3 may be written 8+7.6+3, and 8+s+3 may be written 8+7/6+3.",
"sentence.uniq": "24d9992f3e3e989f138fcb37107e3ccd"
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"937": {
"sentence.id": 937,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the answer to the question is X, we have either (a) I os.",
"sentence.uniq": "5445860b93d403a350dc28b3f764206c"
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"938": {
"sentence.id": 938,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let the answer be X; then 24d.",
"sentence.uniq": "d2ffe49862573d1941afcf94ffbf9df6"
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"939": {
"sentence.id": 939,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let the answer be x; then 24d.",
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"940": {
"sentence.id": 940,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let the answer be X; then 125 c.dm.",
"sentence.uniq": "b33f23a9400f6795d8a629f8e3dcb183"
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"941": {
"sentence.id": 941,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let the answer be x; then 125 =5 x, ..",
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"942": {
"sentence.id": 942,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "To say, for instance, that X is equal to A -B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than",
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"943": {
"sentence.id": 943,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The statement, for instance, that 32 - x = 25, is really a statement that 32 is the sum of x and 25.",
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"944": {
"sentence.id": 944,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From these statements, working backwards, we find successively that v= 5, u = 20, X = 22.",
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"945": {
"sentence.id": 945,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We may therefore conveniently take as our unit, in place of x, a number y such that x=6y.",
"sentence.uniq": "d6b72c8a743ef043da879ba296553431"
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"946": {
"sentence.id": 946,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From this we can deduce successively X - 3s.= 26s., X= 29s.",
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"947": {
"sentence.id": 947,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is arithmetical; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is algebraical, since it involves a change of the number we are thinking",
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"948": {
"sentence.id": 948,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This is a definition of x 3; the sign = is in such cases usually replaced by =.",
"sentence.uniq": "b98a5dda7e2870acac90c24d7cf9a455"
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"949": {
"sentence.id": 949,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the operator 12d X is omitted, the statement is really an equation, giving Is.",
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"950": {
"sentence.id": 950,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We replace 4(x3), for instance, by 4x4.3, because we know that, whatever the value of x may be, the result of subtracting 3 from it and multiplying the remainder by 4 is the same as the result of finding 4x and 4.3 separately and subtracting the latter fr",
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"951": {
"sentence.id": 951,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For multiplication, for instance, we have the statement that, if P and Q are two quantities, containing respectively p and q of a particular unit, then p X Q = q X P; or the more abstract statement that p X q= q X p.",
"sentence.uniq": "54c9e17112d0de828ef0babb06d10217"
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"952": {
"sentence.id": 952,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We cannot solve the equation 7X =4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit w s.",
"sentence.uniq": "860eb694bd82cfd4ddeae378f1d2c903"
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"953": {
"sentence.id": 953,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When, however, we come to the equation x 2 --- 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us.",
"sentence.uniq": "7842217b3720808018eafa0db4e9aa91"
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"954": {
"sentence.id": 954,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definit",
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"955": {
"sentence.id": 955,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We find that the product a m X a m X a m is equal to a im; and, by definition, the product; /a X -la X Ala is equal to a, which is a'.",
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"956": {
"sentence.id": 956,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let X and Y be the related quantities, their expressions in terms of selected units A and B being x and y, so that X=x.A, Y = y.",
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"957": {
"sentence.id": 957,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We take a fixed line OX, usually drawn horizontally; for each value of X we measure a length or abscissa ON equal to x.L, and draw an ordinate NP at right angles to OX and equal to the corresponding value of y .",
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"958": {
"sentence.id": 958,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Y is represented by the length of the ordinate NP, so that the representation is cardinal; but this ordinate really corresponds to the point N, so that the representation of X is ordinal.",
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"959": {
"sentence.id": 959,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the gr",
"sentence.uniq": "a6c79f694d8f498d9bb1d49c1180c5eb"
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"960": {
"sentence.id": 960,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Similarly the equalities 99 X I o I = 9999 = wow - I 98 X 102 = 999 6 = moo() - 4 97 X 10 3 =9991 =1 0000 - 9 lead up to (A - a) (A+a) = A 2 - a 2.",
"sentence.uniq": "c81bd161d72675efff4ee4239fa68267"
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"961": {
"sentence.id": 961,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus, to multiply x 3 -2x+1 by 2x 2 +4, we write the process +I +0-2+I +2+0+4 +2+0-4+2 +0+0 - 0+0 +4+0-8+4 +2+0+0+2-8+4 giving 2x 5 + 2x 2 -8x+4 as the result.",
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"962": {
"sentence.id": 962,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "An equation of the form ax=b, where a and b do not contain x, is the standard form of simple equation.",
"sentence.uniq": "a304918125254ae404625c2069597a26"
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"963": {
"sentence.id": 963,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus 2 x 2 - I - x +x-2 (x _ I) (x+2) is equal to x + 2 q, except when x=1.",
"sentence.uniq": "ee78d123740e2a27acab5c4e92fe7389"
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"964": {
"sentence.id": 964,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It is convenient to retain x, to denote x r /r!, so that we have the consistent notation xr =x r /r!, n (r) =n(r)/r!, n[r] =n[r]/r!.",
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"965": {
"sentence.id": 965,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The binomial theorem for positive integral index may then be written (x + y) n = -iyi +.",
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"966": {
"sentence.id": 966,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we represent this expression by f (x), the expression obtained by changing x into x-+-h is f(x+h); and each term of this may be expanded by the binomial theorem.",
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"967": {
"sentence.id": 967,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we denote these by f i (x), f 2 (x),..",
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"968": {
"sentence.id": 968,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In algebraical transformations, however, such as (x-a)2 = x 2 - 2ax+a 2, the arithmetical rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation.",
"sentence.uniq": "2f444832613864c72a98918bf39b2e6d"
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"969": {
"sentence.id": 969,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus, to divide i by i +x algebraically, we may write it in the form I+o.x+o.x 2 +o.x 3 +o.x 4, and we then obtain I I +0.x+0.x2+0.x3 '+0.x4 = I' x+x2 - x 3 + x4 I+x I+x' where the successive terms of the quotient are obtained by a process which is purely",
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"970": {
"sentence.id": 970,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we divide the sum of x 2 and a 2 by the sum of x and a, we get a quotient x - a and remainder 2a 2, or a quotient a - x and remainder 2x 2, according to the order in which we work.",
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"971": {
"sentence.id": 971,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The highest common factor (or common factor of highest degree) of two rational integral functions of x is therefore found in the same way as the G.C.M.",
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"972": {
"sentence.id": 972,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the same way, to find X I I 4, where X= i+aix+a2x2+.",
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"973": {
"sentence.id": 973,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Considered in this way, the relations between the coefficients of the powers of x in a series may sometimes be expressed by a formal equality involving the series as a whole.",
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"974": {
"sentence.id": 974,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This accounts for the fact that the same table of binomial coefficients serves for the expansions of positive powers of i+x and of negative powers of i - x.",
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"975": {
"sentence.id": 975,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus, to solve the equation ax e +bx+c = o, we consider, not merely the value of x for which ax 2 +bx+c is o, but the value of ax e +bx+c for every possible value of x.",
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"976": {
"sentence.id": 976,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Simultaneous equations in two unknowns x and y may be treated in the same way, except that each equation gives a functional relation between x and y.",
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"977": {
"sentence.id": 977,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We cannot, for instance, say that the fraction C _2 I is arithmetically equal to x+I when x= I, as well as for other values of x; but we can say that the limit of the ratio of x 2 - I to x - I when x becomes indefinitely nearly equal to I is the same as t",
"sentence.uniq": "cab2ccd285d478b6d10729097794237e"
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"978": {
"sentence.id": 978,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We know that log l oN(I+9) = log l oN+log 10 (I+0), and inspection of a table of logarithms shows that, when 0 is small, log 10 (I+B);s approximately equal to X0, where X is a certain constant, whose value is.",
"sentence.uniq": "fdb56d0ddef071210af479e8b2f92fd8"
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"979": {
"sentence.id": 979,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Suppose, for instance, that y=x 2; then to every rational value of x there corresponds a rational value of y, but the converse does not hold.",
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"980": {
"sentence.id": 980,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "They are (a+b)-?-c=a+(b+c) (A) (aXb)Xc=aX(bXc) (A') a+b=b+a (c) aXb=bXa (c') a(b c) =ab-Fac (D) (a - b)+b=a (I) (a=b)Xb=a (I') These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the defini",
"sentence.uniq": "aa03c81f9ac2b97ff1b3d2a8bb8ce2bc"
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"981": {
"sentence.id": 981,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let a+b denote the region made up of a and b together (the common part, if any, being reckoned only once), and let a X b or ab mean the region common to a and b.",
"sentence.uniq": "4e84c03b02a6baebc7881beb95a0e304"
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"982": {
"sentence.id": 982,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For instance, x+y = x+xy and xy = x(x+y) are reciprocal.",
"sentence.uniq": "95b5ea4990adfe1e1e07f1f595f39c45"
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"983": {
"sentence.id": 983,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "So, in general, if we put aA+ 1 3B+yC+...+AL = (a+13+y+...+X)X.",
"sentence.uniq": "2f1ca745c7336bfcdeae87d538340718"
},
"984": {
"sentence.id": 984,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "X is, in general, a determinate point, the barycentre of aA, 3B, &c. (or of A, B, &c. for the weights a, 0, &c.).",
"sentence.uniq": "e97ca3f62f525a98b9eb6ce6ca8048de"
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"985": {
"sentence.id": 985,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "From A Merely Formal Point Of View, We Have In The Barycentric Calculus A Set Of \" Special Symbols Of Quantity \" Or \" Extraordinaries \" A, B, C, &C., Which Combine With Each Other By Means Of Operations And Which Obey The Ordinary Rules, And With Ordinary",
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"986": {
"sentence.id": 986,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 ",
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"987": {
"sentence.id": 987,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "On the other hand, the equations q'x = q and yq' = q have, in general, different solutions.",
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"988": {
"sentence.id": 988,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It is the value of y which is generally denoted by q= q'; a special symbol for x is desirable, but has not been established.",
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"989": {
"sentence.id": 989,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The values of x and y are different, unless V (qq o) = o.",
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"990": {
"sentence.id": 990,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.",
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"991": {
"sentence.id": 991,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "He solved quadratic equations both geometrically and algebraically, and also equations of the form x 2 \"+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.",
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"992": {
"sentence.id": 992,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "His first contention was not disproved until the 15th century, but his second was disposed of by Abul Wefa (940-998), who succeeded in solving the forms x 4 =a and x4-%ax3=b.",
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"993": {
"sentence.id": 993,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Although the foundations of the geometrical resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the equation x 3 = a and x 3 = 2a 3), yet the subsequent development by the Arabs must be",
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"994": {
"sentence.id": 994,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "An imperfect solution of the equation x 3 +-- px 2 was discovered by Nicholas Tartalea (Tartaglia) in 1530, and his pride in this achievement led him into conflict with Floridas, who proclaimed his own knowledge of the form resolved by Ferro.",
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"995": {
"sentence.id": 995,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We imagine a wave-front divided o x Q into elementary rings or zones - often named after Huygens, but better after Fresnelby spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at 0,",
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"996": {
"sentence.id": 996,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus in the case of the circular disk, equidistant (r) from the source of light and from the screen upon which the shadow is observed, the width of the first exterior zone is given by = X(2r)/4(2x), 2x being the diameter of the disk.",
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"997": {
"sentence.id": 997,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by, n, and P (where dS is situated) by x, y, z.",
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"998": {
"sentence.id": 998,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the applications with which we are concerned, t, n are very small quantities; and we may take P = x yn - At the same time dS may be identified with dxdy, and in the de nominator p may be treated as constant and equal to f.",
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"999": {
"sentence.id": 999,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A diminution of X thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to A-2.",
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"1000": {
"sentence.id": 1000,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y.",
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"1001": {
"sentence.id": 1001,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The limits of integration for x may thus be taken to be -2a and -Fla, and for y to be -2b, +2b.",
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"1002": {
"sentence.id": 1002,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Analytically expressed ff+ co x I 2 d dn=ff dxdy= A (9) We have seen that Io (the intensity at the focal point) was equal to A 2 /X 2 f2.",
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"1003": {
"sentence.id": 1003,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If A' be the area over which the intensity must be Io in order to give the actual total intensity in accordance with A'102 =ff + 12d4dn, the relation between A and A' is AA' = X 2 f 2 .",
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"1004": {
"sentence.id": 1004,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = o, and C reduces to C = ff cos px cos gy dx dy,.",
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"1005": {
"sentence.id": 1005,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The extreme discrepancy is that between the waves which travel through the outermost parts of the object-glass at L and L'; so that if we adopt the above standard of resolution, the question is where must P be situated in order that the relative retardati",
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"1006": {
"sentence.id": 1006,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the above argument the whole space between the object and the lens is supposed to be occupied by matter of one refractive index, and X represents the wave-length in this medium of the kind of light employed.",
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"1007": {
"sentence.id": 1007,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The extreme value possible for a is a right angle, so that for the microscopic limit we have Z X o/µ (2).",
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"1008": {
"sentence.id": 1008,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wavelength (X).",
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"1009": {
"sentence.id": 1009,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "No terms in x or x 2 need be considered.",
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"1010": {
"sentence.id": 1010,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The distance f i, which the actual focal length must exceed, is given by d (f1 2 R2) x; so that f1 = 2 R2/X (1) Thus, if X = p j, R= i ?, we find f1= 800 inches.",
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"1011": {
"sentence.id": 1011,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "On either side of any one of them the illumination is distributed according to the same law as for the central image (m = o), vanishing, for example, when the retardation amounts to (mn t 1)X.",
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"1012": {
"sentence.id": 1012,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn+1)X.",
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},
"1013": {
"sentence.id": 1013,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The aperture of the unretarded beam may thus be taken to be limited by x = - h, x = o, y= - 1, y= +1; and that of the beam retarded by R to be given by x =o, x =h, y = - 1, y = +l.",
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"1014": {
"sentence.id": 1014,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are o and +h.",
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"1015": {
"sentence.id": 1015,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When a, b, X are regarded as constant, the first factor may be omitted, - as indeed should be done for consistency's sake, inasmuch as other factors of the same nature have been omitted already.",
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"1016": {
"sentence.id": 1016,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.",
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"1017": {
"sentence.id": 1017,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "According to this notation, the three equations of motion are dt2 = b2v2E + (a2 - b2) d.s dt =b2v2rj+(a2 - b2) dy d2 CIF - b2p2+(a2_b2)dz It is to be observed that denotes the dilatation of volume of the element situated at (x, y, z).",
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"1018": {
"sentence.id": 1018,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Since the dimensions of T are supposed to be very small in com d parison with X, the factor dy (--) is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write TZ y d e T ?",
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"1019": {
"sentence.id": 1019,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave = sin (nt - kx) (19) is supposed to be broken up in passing the plane x = o.",
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"1020": {
"sentence.id": 1020,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Note that these phagocytic cells are pushing out protoplasmic processes (pseudopodia) by which they grasp their victims. (X woo diam.) FIG.",
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},
"1021": {
"sentence.id": 1021,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "These vesicles are filled with the colloid material (x 90 diam.) J FIG.",
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"1022": {
"sentence.id": 1022,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Brussels was again the headquarters in 1741, by which time Voltaire had finished the best and the second X XVIII.",
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"1023": {
"sentence.id": 1023,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It is not probable that the sweet-smelling gums and resins of the countries of the Indian Ocean began to be introduced into Greece before the 8th or 7th century B.C., and doubtless XiOavos or X q /3avw-rOs first became an article of extensive commerce onl",
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"1024": {
"sentence.id": 1024,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The beaches which had been selected were, enumerating from right to left, \" S \" in Morto Bay, \" V \" and \" W \" on either side of Cape Helles at the south-western end, and \" X \" and \" Y \" on the outer shore; \" V \" and \" W \" were regarded as of primary impor",
"sentence.uniq": "bfdf55ee49672fdf6912e1160627e78c"
},
"1025": {
"sentence.id": 1025,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The attacks at \" S \" and \" Y \" were intended to be subsidiary; but great importance was attached to that at \" X \" owing to the vicinity of this point to \" W.\"",
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},
"1026": {
"sentence.id": 1026,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "As it turned out, the actual disembarkations at \" S,\" \" X \" and \" Y \" were carried out without any very great difficulty; but the troops detailed for \" W \" beach only gained a footing after incurring very heavy losses and by a display of indomitable resol",
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"1027": {
"sentence.id": 1027,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "But the forces which had landed at \" W \" and \" X \" beaches had joined hands, the one battalion detailed for \" S \" beach had secured a good position, and during the night the troops still left aboard the \" River Clyde \" contrived to disembark.",
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},
"1028": {
"sentence.id": 1028,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Additional infantry was got ashore at \" W \" and \" X \" beaches, the first elements of the French division began disembarking at \" V \" beach in the afternoon, and before evening touch had been gained with the battalion that had made good at \" S \" beach.",
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"1029": {
"sentence.id": 1029,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "These were \"lost sheep of the house of Israel\"; but Christ's freedom from Jewish exclusiveness is also brought out (I) as regards Samaritans, by the rebuke administered to the disciples at ix.52 sqq., the parable in x.",
"sentence.uniq": "cea59afcb1918f0826a2814627ddbac7"
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"1030": {
"sentence.id": 1030,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the second of these passages the disciples are exhorted to choose a life of voluntary poverty; the nearest parallel is the ideal set before the rich young man at Mark x.",
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"1031": {
"sentence.id": 1031,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The cartesian equation referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic equation is s = c tan The radius of curvat",
"sentence.uniq": "d9b66139ec8e412041ad0b02a5d5afb4"
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"1032": {
"sentence.id": 1032,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus water being about Boo times denser than air and mercury 13.6 times denser than water, k/h = 6,/p = 800 X 13.6 = Io,880; (2) and with an average barometer height of 30 in.",
"sentence.uniq": "91fae657a8278c78e47cd69716b83624"
},
"1033": {
"sentence.id": 1033,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the part",
"sentence.uniq": "1c03607c23a28b414935c3ef0e5626ce"
},
"1034": {
"sentence.id": 1034,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus if the plane is normal to Or, the resultant thrust R =f fpdxdy, (r) and the co-ordinates x, y of the C.P. are given by xR = f f xpdxdy, yR = f f ypdxdy.",
"sentence.uniq": "86e291cc6d54c9bd803647121cc405f1"
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"1035": {
"sentence.id": 1035,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, co",
"sentence.uniq": "c27e1119fbee77c66c95a69b5c7bbbcc"
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"1036": {
"sentence.id": 1036,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = o, = o; and in steady motion the equations reduce to dH/dv=2q-2wn = 2gco sin e, (4) where B is the angle between the stream line and vortex line;",
"sentence.uniq": "af62e08052a6b2fff345c0307c28c389"
},
"1037": {
"sentence.id": 1037,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the equations of uniplanar motion = dx - du = dx + dy = -v 2 ?, suppose, so that in steady motion dx I +v24 ' x = ?'",
"sentence.uniq": "a620860d1e9b76e6e661f1d9481f66d7"
},
"1038": {
"sentence.id": 1038,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 a",
"sentence.uniq": "2b5d7f7d728ef597136f9ae58c9d4aa7"
},
"1039": {
"sentence.id": 1039,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Then, if the outside cylinder is free to move - 2a2 2 X 1 = 0, T T = b2 a2, 7rpa 2 Ub 2+ a 2.",
"sentence.uniq": "7a67d8261f91927ec15176d07ecda1be"
},
"1040": {
"sentence.id": 1040,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is p",
"sentence.uniq": "1fea4ffb00420aa0fef2779916359b9c"
},
"1041": {
"sentence.id": 1041,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For the liquid filling the interior of a rotating elliptic cylinder of cross section x2/a2+y2/b2 = 1, /(4) = m l (x 2 / a2 - - y2 /b 2) (5) with V21G1' =-2R =-2 m 1 (I / a2 + 21b2), 214 = m l (x2 / a2 + y2 / b2) - IR(x2+y2) = I R (x2 - y2) (a 2 - b2)/(a2+",
"sentence.uniq": "1782fdcfd0ccdebc2cfbfc21762ba850"
},
"1042": {
"sentence.id": 1042,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The polar equation of the cross-section being rI cos 19 =al, or r + x = 2a, (3) the conditions are satisfied by = Ur sin g -2Uairi sin IB = 2Uri sin 10(14 cos 18a'), (4) 1J/ =2Uairi sin IO = -U1/ [2a(r-x)], (5) w =-2Uaiz1, (6) and the resistance of the li",
"sentence.uniq": "badc764dcf4d00de791c02dccfb7baa1"
},
"1043": {
"sentence.id": 1043,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A relative stream line, along which 1/,' = Uc, is the quartic curve y-c=?![2a(r-x)], x = 4a2y2-(y g)4, r- 4a2y2 +(y c) 4, 7) 4 a (y-c) 4a(y and in the absolute space curve given by 1', dy= (y- c)2, x= 2ac_ 2a log (y -c) (8) 2ay y - c 34.",
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"1044": {
"sentence.id": 1044,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar coordinates, x =r cos 0, y = r sin 0, the equation (2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+)",
"sentence.uniq": "d538f9c5fa314c0aff2425f1d6abaaf7"
},
"1045": {
"sentence.id": 1045,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relati",
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"1046": {
"sentence.id": 1046,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about th",
"sentence.uniq": "75592a719d174c145e7f42689512bf41"
},
"1047": {
"sentence.id": 1047,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If there are more B corners than one, either on xA or x'A', the expression for i is the product of corresponding factors, such as in (5) Restricting the attention to a single corner B, a = n(cos no +i sin 110) _ (b-a'.0-a) +1!",
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"sentence.sentence": "Ja - u ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx ",
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"sentence.sentence": "Generally, by making a' = -oo, the line x'A' may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplica tion the result can be ob A tained, with assigned n, a, and b, of the efflux from a symmetrical conver",
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"sentence.sentence": "For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only, 41 = - x, 42 = -, 43 = - z; (2) but a rotation will stir up the liquid in the cavity, so that the'x's depend on the shape o",
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"sentence.sentence": "Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, px _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p",
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"sentence.sentence": "The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular are",
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"sentence.sentence": "When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the f",
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"sentence.sentence": "Between two concentric spheres, with a 2 +A = r 2, a2+A1=a12, A=B =C =a3/3r3, a 3 a 3 a3 _ a3 Cb _1U x r 3 a13 Y'=2 Uy2 r3 3 a13 .",
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"sentence.sentence": "The extension to the case where the liquid is bounded externally by a fixed ellipsoid X= X is made in a similar manner, by putting 4 = x y (x+ 11), (io) and the ratio of the effective angular inertia in (9) is changed to 2 (B0-A0) (B 1A1) +.a12 - a 2 +b 2",
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"sentence.sentence": "The determination of the O's and x's is a kinematical problem, solved as yet only for a few cases, such as those discussed above.",
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"sentence.sentence": "Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.",
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"sentence.sentence": "Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are dT dT dT (I) = dU + x2=dV, x3 =dW, dT dT dT Yi dp' dQ' y3=dR; but when it is expressed as a quadratic function of xi, 'x2, x3, yi, Y2, Y3, U =",
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"sentence.sentence": "When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.) T =constant, x 2 +x 2 +x 2 =F 2, a constant, (iii.) x1y1 +x2y2+x3y3 =n = GF, a constant; and the dynamical equations in (3) express th",
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"sentence.sentence": "If p denotes the density of the air or medium W' = sird 3 xp, (23) W' I p __ W I -1 3 k12 I k22 x2 ±i a 2= 101-1 3 '111 2= 2 tan g S = Q (l - a) x 2+ I (26) in which a/p may be replaced by 800 times the S.G.",
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"sentence.sentence": "Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving S the Angle of Rifling, and n the Pitch of Rifling in Calibres.",
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"sentence.sentence": "The components of force, X, Y, and N, acting on the liquid at 0, and reacting on the body, are then X=It.",
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"sentence.sentence": "The Book of Noah is mentioned in Jubilees x.",
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"sentence.sentence": "As for the author, he was no Essene, for he recognizes animal sacrifices and cherishes the Messianic hope; he was not a Sadducee, for he looks forward to the establishment of the Messianic kingdom (x.); nor a Zealot, for the quietistic ideal is upheld (ix",
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"sentence.sentence": "B, Archigetes sieboldi; X 60 (from Leuckart).",
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"sentence.sentence": "The uterus (X in figure C) begins in all cases at the shell gland (c, d) and may exhibit a swelling (R S) for the retention of the spermatozoa..",
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"sentence.sentence": "C, invaginated head of Cysticercus cellulosae, showing the bent neck and receptacle r; X 30.",
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"sentence.sentence": "Zinc carbonate, ZnCO 3, occurs in nature as the mineral calamine (q.v.), but has never been prepared artificially, basic carbonates, ZnCO 3 .xZn(OH) 2, where x is variable, being obtained by precipitating a solution of the sulphate or chloride with sodium",
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"sentence.sentence": "A queen of this people (the \" Queen of Sheba \") is said (1 Kings x.) to have visited Solomon about 950 B.C. There is, however, no mention of such a queen in the inscriptions.",
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"sentence.sentence": "Addai was supposed to be one of the Seventy of Luke x.",
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"sentence.sentence": "This functionary is first formally mentioned under Leo X.(1513-1521) in the proceedings in connexion with the canonization of St Lorenzo Giustiniani.",
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"sentence.sentence": "A, Fasciola hepatica, from the ventral surface (X 2); the alimentary and nervous systems only shown on the left side of the figure, the excretory only on the right; a, right main branch of the intestine; c, a diverticulum; g, lateral ganglion; n, lateral ",
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"sentence.sentence": "The central nervous system (x) is highly developed, and in Loxosoma bears a pair o` eyes.",
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"sentence.sentence": "In its modern form the theorem, which is true for all values of n, is written as (x +a) n -1+ I.",
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"sentence.sentence": "Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms R- x 3 3x 3, 3x 3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x--",
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"sentence.sentence": "I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this s",
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"sentence.sentence": "The hyperbolic or Gudermannian amplitude of the quantity x is ta n (sinh x).",
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"sentence.sentence": "Hence the density v is given by 47rabc (x2/a4+y2/b4-I-z2/c4), and the potential at the centre of the ellipsoid, and therefore its potential as a whole is given by the expression, adS Q dS V f r 47rabc r' (x2/a4-I-y2/b4+z2/c4) Accordingly the capacity C of",
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"sentence.sentence": "It has been shown above that the potential due to a charge of q units placed on a very small sphere, commonly called a point-charge, at any distance x is q/x.",
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"sentence.sentence": "The mathematical importance of this function called the potential is that it is a scalar quantity, and the potential at any point due to any number of point charges ql, q2, q3, &c., distributed in any manner, is the sum of them separately, or qi/xl+q2/x2+",
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"sentence.sentence": "Then the electric force due to the point s' charge q at distance x is q/x, and the resolved part normal to the element of surface dS is q cos0/x 2.",
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"sentence.sentence": "Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).",
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"sentence.sentence": "Its angular points have then co-ordinates (x t Zdx, y t Zdy, z * zdz).",
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"sentence.sentence": "Let V be the potential at the centre of the prism, then the normal forces on the two faces of area dy.dx are respectively RI dx2 d xl and (dx 2 d x), dV d2 and similar expressions for the normal forces to the other pairs of faces dx.dy, dz.dx.",
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"sentence.sentence": "See La Fondation de la regence d'Alger, histoire des Barberousse, chronique arabe du X VI siecle published by Sander Rang and Ferdinand Denis, Paris, 1837 - for a curious Moslem version of their story.",
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"sentence.sentence": "I, 4), the modern Citta di Castello, he set up a temple at his own expense and adorned it with statues of Nerva and Trajan (x.",
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"sentence.sentence": "The width of each of the portions aghc and acfe cut away from the lens was made slightly greater than the focal length of lens X tangent of sun's greatest diameter.",
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"sentence.sentence": "Here, in order to fulfil the purposes of the previous models, the distance of the centres of the lenses from each other should only slightly exceed the tangent of sun's diameter X focal length of lenses.",
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"sentence.sentence": "Thirdly, the vials source from the time of Pompey (circa 63),- x.",
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"sentence.sentence": "The author wrote x.",
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"sentence.sentence": "In it the prophet receives a new commission, x.",
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"sentence.sentence": "Buchanan found a mean of 20 experiments made by piezometers sunk in great depths on board the \" Challenger \" give a coefficient of compressibility K=491 X 107; but six of these experiments made at depths of from 2740 to 3125 fathoms gave K=480Xio 7.",
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"sentence.sentence": "Feuillet des Conches Louis X VI., Marie Antoinette et Madame Elisabeth, lettres et documents inedits (6 vols., Paris, 1864-1873), while most of the works on Marie Antoinette published before the appearance of Arneth's publications (1865, &c.) are based pa",
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"sentence.sentence": "The radial distance x is at any instant proportional to the force acting through the spring.",
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"1098": {
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"sentence.sentence": "It appears tolerably safe to conclude that, whatever errors 'may have affected the determination, the diameter or distance of the particles of water is between the two thousand and the ten thousand millionth of an inch \" (= between 125 X I o 8 and 025 X 1",
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"1099": {
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"sentence.sentence": "Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecul",
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"1100": {
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"sentence.sentence": "T =273, we obtain R =1.35 X Io -16 and this enables us to determine the mean velocities produced by heat motion in molecules of any given mass.",
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"1101": {
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"sentence.sentence": "Plato (Rep. x.) embodies the idea in one of his finest myths.",
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"sentence.sentence": "The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.",
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"sentence.sentence": "The finest of these apartments, containing beautiful arabesque x XVII.",
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"sentence.sentence": "Take any base X'X, and draw lines at right angles to this base through all the angular points of the figure.",
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"sentence.sentence": "Then, if we take ordinates Kb, Lg, Mc, Nd, Pf, equal to B'B, GG', C'C, D'D, FF', the figure abgcdfe will be the equivalent trapezoid, and any ordinate drawn from the base to the a LM N P e X top of this trapezoid will be equal to the portion of this ordin",
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"sentence.sentence": "Volume =height X area of end =length of edge X area of cross-section; the \" height \" being the perpendicular distance between the two ends.",
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"sentence.sentence": "B where x = the distance between the two lines = N /No.",
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"1108": {
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"sentence.sentence": "These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.",
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"1109": {
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"sentence.sentence": "Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.",
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"1110": {
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"sentence.sentence": "Let KA and LB be the positions of NP corresponding to the extreme values of x.",
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"sentence.sentence": "To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.",
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"1112": {
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"sentence.sentence": "It arises mainly in statistics, when the ordinate of the trapezette represents the relative frequency of occurrence of the magnitude represented by the abscissa x; the magnitude of the abscissa corresponding to the median ordinate is then the \" median val",
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"sentence.id": 1113,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If there are m of these strips, and if the breadth of each is h, so that H =mh, it is convenient to write x in the form xo+Oh, and to denote it by x 0, the corresponding value of u being ue.",
"sentence.uniq": "1061a1827cee6c9aa36847894195323c"
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"1114": {
"sentence.id": 1114,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the case of the briquette the position of the foot of the ordinate u is expressed by co-ordinates x, y, referred to a pair of axes parallel to a pair of sides of the base of the briquette.",
"sentence.uniq": "460c891a63ad98c8f5c08d54dd3c5279"
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"1115": {
"sentence.id": 1115,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the planes of one set divide it into m slabs of thickness h, and those of the other into n slabs of thickness k, so that H =mh, K = nk, then the values of x and of y for any ordinate may be denoted by xo+Oh and yo+Ok, and the length of the ordinate by ",
"sentence.uniq": "27994f3b94d3eb6b3f41482091224079"
},
"1116": {
"sentence.id": 1116,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The area of the trapezette, measured from the lower bounding ordinate up to the ordinate corresponding to any value of x, is some function of x.",
"sentence.uniq": "2e2a87d823983f34674c362f94b57d05"
},
"1117": {
"sentence.id": 1117,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the notation of the integral calculus, this area is equal to f x o udx; but the notation is inconvenient, since it implies a division into infinitesimal elements, which is not essential to the idea of an area.",
"sentence.uniq": "23fad15c9bc5d5df86c29ba6763f0534"
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"1118": {
"sentence.id": 1118,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the same way the volume of a briquette between the planes x = xo, y = yo, x= a, y = b may be denoted by [[Vx,y ]y=yo] u 'x' =xo.",
"sentence.uniq": "be8738b53027fdccd5d83511c515b163"
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"1119": {
"sentence.id": 1119,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The statement that the ordinate u of a trapezette is a function of the abscissa x, or that u=f(x), must be distinguished from u =f(x) as the equation to the top of the trapezette.",
"sentence.uniq": "bbbb0c91e2c870ce9231b82d58d0cfa6"
},
"1120": {
"sentence.id": 1120,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If this is done for every possible value of x, there will be a series of ordinates tracing out a trapezette with base along OX.",
"sentence.uniq": "b4d01389d66709358b13cc0907262a38"
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"1121": {
"sentence.id": 1121,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Hence the volume of each element of the solid figure is to be found by multiplying the area of the corresponding element of the trapezette by 1, and therefore the total volume is 1 X area of trapezette.",
"sentence.uniq": "7d636fc4c2264debb02356742c533462"
},
"1122": {
"sentence.id": 1122,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The simplest case is that in which u is constant or is a linear function of x, i.e.",
"sentence.uniq": "4cde20bca65f27a35f599f1ed826cbac"
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"1123": {
"sentence.id": 1123,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The next case is that in which u is a quadratic function of x, i.e.",
"sentence.uniq": "d4e926092e599adcf005b66df28d0563"
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"1124": {
"sentence.id": 1124,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Generally, if the area of a trapezette for which u is an algebraical function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapeze",
"sentence.uniq": "b39e774321e3c6f10e6ea79388d3e0a1"
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"1125": {
"sentence.id": 1125,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This will be seen by taking the mid-ordinate as the ordinate for which x = o, and noticing that the odd powers of x introduce positive and negative terms which balance one another when the whole area is taken into account.",
"sentence.uniq": "05ed5495eeb387573f0f73d3f1f3aa26"
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"1126": {
"sentence.id": 1126,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "When u is of degree 4 or 5 in x, we require at least five ordinates.",
"sentence.uniq": "96d40739f5fe6e59fe5f0c43271cdd0c"
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"1127": {
"sentence.id": 1127,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Writing m = 2p, and grouping the coefficients of the successive differences, we shall find area = 2ph up+ 2 652up + 3 p4365p2 84up 3p,6 - 21p4 28p2 15120 If u is of degree 2f or 2f + i in x, we require to go up to b 2f u p, so that m must be not less than",
"sentence.uniq": "a4d7c66764f4b1a7c8ab6580e24e667b"
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"1128": {
"sentence.id": 1128,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, wher",
"sentence.uniq": "b629446582d107ce895da80a224b5221"
},
"1129": {
"sentence.id": 1129,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the case of the sphere, for instance, whose radius is R, the area of the section at distance x from the centre is lr(R 2 -x 2), which is a quadratic function of x; the values of So, Si, and S2 are respectively o, 7rR 2, and o, and the volume is therefo",
"sentence.uniq": "484047ff3d0764cc6ceffeb25632e71a"
},
"1130": {
"sentence.id": 1130,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the case of a pyramid, of height h, the area of the section by a plane parallel to the base and at distance x from the vertex is clearly x 2 /h 2 X area of base.",
"sentence.uniq": "865b229d1edfad59d5637f56dd3bff23"
},
"1131": {
"sentence.id": 1131,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Let the distance between the parallel planes through AB and CD be h, and let a plane at distance x from the plane through AB cut the edges AC, up -f- .",
"sentence.uniq": "375c4827e8406b696a713b5d2b7b1af7"
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"1132": {
"sentence.id": 1132,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "By drawing Ac and Ad parallel to BC and BD, so as to meet the plane through CD in c and d, and producing QP and RS to meet Ac and Ad in q and r, we see that the area of Pqrs is (x/h - x 2 /h 2) X area of cCDd; this also is a quadratic function of x.",
"sentence.uniq": "f0862be95dde1e211ff9c6811d7f7fe9"
},
"1133": {
"sentence.id": 1133,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the case of the parabolic trapezette, for instance, xu is of degree 3 in x, and therefore the first moment is lh(xouo+4xlui-+x2u2).",
"sentence.uniq": "d6ad967426f053d56f5bf7bc00557d67"
},
"1134": {
"sentence.id": 1134,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of mas",
"sentence.uniq": "fe6a56af5b6a311bcc01cd5af22df6c6"
},
"1135": {
"sentence.id": 1135,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..",
"sentence.uniq": "c32da55ad783bae4b6d13fa637b6300b"
},
"1136": {
"sentence.id": 1136,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "To extend these methods to a briquette, where the ordinate u is an algebraical function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in t",
"sentence.uniq": "48666c2aad7c2da55eec619d663f48b7"
},
"1137": {
"sentence.id": 1137,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This ordinate will be an algebraical function of x, and we can again apply a suitable formula.",
"sentence.uniq": "78a424275f63450e2431d22d3b1de749"
},
"1138": {
"sentence.id": 1138,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Suppose, for instance, that u is of degree not exceeding 3 in x, and of degree not exceeding 3 in y, that it contains terms in x3y3, x 3 y 2, x2y3, &c.; and suppose that the edges parallel to which x and y are measured are of lengths 2h and 3k, the brique",
"sentence.uniq": "729648223079a3d36ad81a5980defede"
},
"1139": {
"sentence.id": 1139,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The area of the section by a plane at distance x from the edge 0 is a function of x whose degree is the same as that of u.",
"sentence.uniq": "14126806f014cc6e6ff7a3225f4b1a3c"
},
"1140": {
"sentence.id": 1140,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The volume of the briquette for which u is a function of x and y is found by the operation of double integration, consisting of two successive operations, one being with regard to x, and the other with regard to y; and these operations may (in the cases w",
"sentence.uniq": "24798469be3c24731c0fda4130500f3d"
},
"1141": {
"sentence.id": 1141,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an algebraical function of x or of x and y, and expressing the area or volume of this latter ",
"sentence.uniq": "9a91cff9117d6240115d33ef67a67387"
},
"1142": {
"sentence.id": 1142,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This means that, if the minor trapezette consists of k strips, v will be of degree k or k - I in x, according as the data are the bounding ordinates or the mid-ordinates.",
"sentence.uniq": "9e34f72961312bbbcf27f9c18213326c"
},
"1143": {
"sentence.id": 1143,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "P2 = x2 I A I + x 2 I A I +.",
"sentence.uniq": "9f3be38bba88edfaa29471e82d1d6ced"
},
"1144": {
"sentence.id": 1144,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The position of the central ordinate is given by x = v 1 /po, and therefore is given approximately by pl/po.",
"sentence.uniq": "fe95e2792a8ac34c280b9fbea2687b5d"
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"1145": {
"sentence.id": 1145,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "These results may be extended to the calculation of an expression of the form fxo u4(x)dx, where 0(x) is a definite function of x, and the conditions with regard to u are the same as in § 82.",
"sentence.uniq": "c50d0853aad5fb4e0592cf9d9f9ae777"
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"1146": {
"sentence.id": 1146,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The formulae can be adapted to the case in which cp(x) is tabulated for xl,.",
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"1147": {
"sentence.id": 1147,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The expressions in square brackets are in each case to be taken as relating to the extreme values x =xo and x=xm, as in §§ 75 and 76.",
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},
"1148": {
"sentence.id": 1148,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x =x0, x = x i,",
"sentence.uniq": "33702b1b2c5775f896c68920e28a5e8a"
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"1149": {
"sentence.id": 1149,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the data of the briquette are, as in § 86, the volumes of the minor briquettes, but the condition as to close contact is not satisfied, we have y \"`x P u dx dy = K + L + R - X111010-0,0 f xo yo i'?",
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"1150": {
"sentence.id": 1150,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Either or both of the expressions K and L will have to be calculated by means of the formula of § 84; if this is applied to both expressions, we have a formula which may be written in a more general form f f 4 u4(x, y) dx dy = u dx dy.",
"sentence.uniq": "41ce46ebc0e79213b64276c1d726b48b"
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"1151": {
"sentence.id": 1151,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It is also clearly impossible to express u as an algebraical function of x and y if some value of du/dx or duldy is to be infinite.",
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"1152": {
"sentence.id": 1152,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "One method is to construct a table for interpolation of x in terms of u, and from this table to calculate values of x corresponding to values of u, proceeding by equal intervals; a quadrature-formula can then be applied.",
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"1153": {
"sentence.id": 1153,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.",
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"1154": {
"sentence.id": 1154,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from 'a known value of X udx.",
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"1155": {
"sentence.id": 1155,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hdv/dx or h 2 d 2 v /dx 2 can be conveniently expressed in terms of v and x to a greater degree of accuracy than it could be found from ",
"sentence.uniq": "83fec2e12a6c13f4a9de0de33e032bd7"
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"1156": {
"sentence.id": 1156,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "As Johnson thought it unsafe to pursue the routed army his victory had no other effect than the erection here of the useless defences of Fort William Henry, but as it was the only success in a year of gloom parliament rewarded him with a grant of X 5000 a",
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"1157": {
"sentence.id": 1157,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Both narratives are doubtless based upon actual occurrences - the cures narrated in Mark ii., iii., viii., x.",
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"1158": {
"sentence.id": 1158,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Mary and Martha are admittedly identical with the sisters in Luke x.",
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"1159": {
"sentence.id": 1159,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus a push or a compression of the X air is transmitted onwards in the direction OX.",
"sentence.uniq": "cea1a1e8dde0371161a4aed566b696f0"
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"1160": {
"sentence.id": 1160,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "There is also the \" external \" applied pressure X, and the total momentum flowing out per second is X-I-P4-W-1-p(U - u)2.",
"sentence.uniq": "1cc1aede45881190911a17b3c21e867f"
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"1161": {
"sentence.id": 1161,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space.",
"sentence.uniq": "bdce974c9801cad34945cb9f41b01528"
},
"1162": {
"sentence.id": 1162,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity U if we apply the pressure X at every point of the disturbance.",
"sentence.uniq": "98165e644d194210597cba42d1810599"
},
"1163": {
"sentence.id": 1163,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the velocity U is so chosen that E - poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).",
"sentence.uniq": "43f9a06cfb6810cb742c464487c750c6"
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"1164": {
"sentence.id": 1164,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propa",
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"1165": {
"sentence.id": 1165,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the momentum equation (4) we may now omit X and it becomes 0.+P(U - u) 2 =poU2.",
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"1166": {
"sentence.id": 1166,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Thus the second A = 2 X 2 X 2, and we may regard it as an ascent through two fifths in succession and then a descent through an octave.",
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"1167": {
"sentence.id": 1167,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The third 4= 5 X Z X z or ascent through an interval 4, which has no special name, and a descent through two octaves, and so on.",
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"1168": {
"sentence.id": 1168,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we start with F as key-note, besides a small difference at we have as the fourth from it 3 X 4 = y, making with B = I R 5 an interval and requiring a new note, B flat.",
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"1169": {
"sentence.id": 1169,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Since U=n X where U is the velocity of sound, X the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.",
"sentence.uniq": "507e31c0cef7dcd03b462212f4cec073"
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"1170": {
"sentence.id": 1170,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The simplest form of wave, so far as our sensation goes - that is, the one giving rise to a pure tone - is, we have every reason to suppose, one in which the displacement is represented by a harmonic curve or a curve of sines, y=a sin m(x - e).",
"sentence.uniq": "9e39d3b7150aa45b713369d860171523"
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"1171": {
"sentence.id": 1171,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we put this in the form y=a sin (x - e), we see that y=o, for x =e, e +aX, e+2A, e+;X, and so on, that y is + from x=e to x=e+iX, - from e+zX to e+?A, and so on, and that it alternates between the values+a and - a.",
"sentence.uniq": "38a6206f89153d40cf117ffe715f0e77"
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"1172": {
"sentence.id": 1172,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If we transfer 0 to A, e=o, and the curve may be represented by y=a sin A x.",
"sentence.uniq": "feb3457e05d5e54a2a19fc4bc35f48aa"
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"1173": {
"sentence.id": 1173,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).",
"sentence.uniq": "3caffef168f446a55e9e530fdce59f01"
},
"1174": {
"sentence.id": 1174,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...",
"sentence.uniq": "edea9ef78011f1741d7a58fa0da0ac99"
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"1175": {
"sentence.id": 1175,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fund",
"sentence.uniq": "fccc34d70d8bf00befe7490622b29cce"
},
"1176": {
"sentence.id": 1176,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Putting A /M =n 2 the equation becomes x+n 2 x=o, whence x =A sin nt, and the period is 27r/n.",
"sentence.uniq": "4f04a418796e7fd29dbd59c5a637ddc2"
},
"1177": {
"sentence.id": 1177,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The result was an amplitude of 1.27 X 107 cm.",
"sentence.uniq": "78adc37027b051b68bbe5a0de10faaaa"
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"1178": {
"sentence.id": 1178,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We may represent the displacement due to one of the trains by y l =a sin 2 i (24) where x is measured as in equation (16) from an ascending node as A in fig.",
"sentence.uniq": "bf69c53b77644a009f72f0ac2b2df0e1"
},
"1179": {
"sentence.id": 1179,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If w is the total pressure excess, and if y is the total displacement at x, then w = E Xchange of volume _original volume = - Edy/dx.",
"sentence.uniq": "69920dcce133c2a31975fdae74a14a78"
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"1180": {
"sentence.id": 1180,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If U is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.",
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"1181": {
"sentence.id": 1181,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We might make this investigation more general by introducing a force X as in the investigation for air, but it hardly appears necessary.",
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"1182": {
"sentence.id": 1182,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The middle of the string is a loop. In the next mode A and C represent the ends and AC = X = and n 2 = U/A 2 = 2U/2/ = (2/21) A / (T/m).",
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"1183": {
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"sentence.sentence": "If then y=a sin (x - Ut),, t=2,ro-ra cos (x - Ut).",
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"1184": {
"sentence.id": 1184,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types - fixed at one end, free at both ends, fixed at both",
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"1185": {
"sentence.id": 1185,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "If the two interfering waves, being still of same length X, be in opposite phases, or sõ that one is in advance of the other by 2X, and consequently one produces in the air the opposite state of motion to the other, then the resultant wave is one of the s",
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"1186": {
"sentence.id": 1186,
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"sentence.sentence": "If the difference of phase be varied gradually from zero to - X, the resulting sound will 2 gradually decrease from a maximum to a minimum.",
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"1187": {
"sentence.id": 1187,
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"sentence.sentence": "The amplitudes of these tones are proportional to the products of a and b multiplied by X or µ.",
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"1188": {
"sentence.id": 1188,
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"sentence.sentence": "But inasmuch as the successive orders are proportional to A X 2 A 3, or µµ 2 µ 3, and X and µ are small, they are of rapidly decreasing importance, and it is not certain that any beyond those in equation (35) correspond to our actual sensations.",
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"1189": {
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"sentence.sentence": "We may see how this occurs by supposing that the restoring force of the receiving mechanism is represented by Ax- f-µx 2, where x is the displacement and µx 2 is very small.",
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"1190": {
"sentence.id": 1190,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Now µx 2 is very small compared with Ax, so that x is nearly equal to F/X, and as an approximation, F=Ax+µF 2 /A 2, or x=F/A - µF 2 /A 3.",
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"1191": {
"sentence.id": 1191,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In connexion with the journey from this region to Jerusalem three striking incidents are recorded, x.",
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"1192": {
"sentence.id": 1192,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In one is represented Moses receiving the Old Law, in the other Christ delivers to St Peter the New Law - a charter sealed with the X P monogram.",
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"1193": {
"sentence.id": 1193,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.",
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"1194": {
"sentence.id": 1194,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of i",
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"1195": {
"sentence.id": 1195,
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"sentence.sentence": "Maitland describes it (Political Theories of the Middle Ages, p. x.), have an essentially \"statelike character.\"",
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"1196": {
"sentence.id": 1196,
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"sentence.sentence": "Let a b be Ak X ?",
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"1197": {
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"sentence.sentence": "The greatest moment CG at C is x(l-x)/l.",
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"sentence.sentence": "Hence the moment of the load on Am at C is wy0m, and the moment of a uniform load over any portion of the girder is w X the area of the influence curve Ip' G' E ' under that portion.",
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"1199": {
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"sentence.sentence": "If X ---- r - i the scales are so chosen i that a inch represents 1 x ' n - \"--, in.",
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"1200": {
"sentence.id": 1200,
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"sentence.sentence": "If the load rests directly on the main girder, the greatest -Iand - shears at C will be wXAGC and -w X CHB.",
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"1201": {
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"sentence.sentence": "Similarly, rs = W (x +a) 11 is the shear on DB.",
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"1202": {
"sentence.id": 1202,
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"sentence.sentence": "H = wL2/8y, or, calling x the distance from the vertex to the point of support, H = wx2/2y.",
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"1203": {
"sentence.id": 1203,
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"sentence.sentence": "The value of R, the tension at any point at a distance x from the vertex, is obtained from the equation R 2 = H2 +V2 = w2x4 /4Y 2 +w2x2, or, 2.",
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"1204": {
"sentence.id": 1204,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "R=wx (I+x2/4Y2) Let i be the angle between the tangent at any point having the co-ordinates x and y measured from the vertex, then 3..",
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"1205": {
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"sentence.sentence": "The integration can be performed when M is expressed in terms of x.",
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"1206": {
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"sentence.sentence": "Thus for a beam supported at the ends and loaded with w per inch length M =w (a 2 - x 2), where a is the half span.",
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"1207": {
"sentence.id": 1207,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Then the deflection at the centre is the value of y for x = a, and is _ 5 wa4 S - 14 EI' The radius of curvature of the beam at D is given by the relation R=EI/M.",
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"1208": {
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"sentence.sentence": "They pay no premium, and generally receive a salary of P8 to £ 12 in the first year, rising annually to X30 or £35 as staff nurse, and subsequently to x;40 or £50 as sister or head nurse.",
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"1209": {
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"sentence.sentence": "In the spectroscope calcium exhibits two intense lines-an orange line (a), (X 6163), a green line (a), (X 4229), and a fainter indigo line.",
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"1210": {
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"sentence.sentence": "Potassium and sodium readily dissolve in the anhydrous acid with evolution of hydrogen and formation of x.",
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"1211": {
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"sentence.sentence": "Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos\" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the ",
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"sentence.sentence": "We shall make the natural supposition that motion of the aether, say with velocity (u,v,w) at the point (x,y,z), is simply superposed on the velocity V of the optical undulations through that medium, the latter not being intrinsically altered.",
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"1213": {
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"sentence.sentence": "We may now, as is somewhat the more natural course in the terrestrial application, take axes (x,y,z) which move with the matter; but the current must be invariably defined by the flux across surfaces fixed in space, so that we may say that relation (i) re",
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"1214": {
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"sentence.sentence": "Trains of waves nearly but not quite homogeneous as regards wave-length will as usual be propagated as wave-groups travelling with the slightly different velocity d(VX-1)/dX-', the value of K occurring in V being a function of X determined by the law of o",
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"sentence.sentence": "Acts x.",
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"1216": {
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"sentence.sentence": "Such in the main remained the situation until 1816, duties being indeed raised from x xv1.",
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"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The equations of motion are now, the co-ordinates x and y being measured in feet, 2 (26) - -rr- - C, dt2 dty - g' * These numbers are taken from a part omitted here of the abridged ballistic table.",
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"1218": {
"sentence.id": 1218,
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"sentence.sentence": "Dividing again by x, as given in (76), tan0.",
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"1219": {
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"sentence.sentence": "The field covers a district about 8 X 10 m.",
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"1220": {
"sentence.id": 1220,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "We have such analytical formulae as that is, any number from I to 15 can be made with the parts I, 2, 4, 8; and similarly any can be made up, and in one way only, with A like formula is I - x 3 I - x9 I - x27 I x.",
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"1221": {
"sentence.id": 1221,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "In the oceans alone there are estimated to be 1141 X 10 12 tons of sulphate, K 2 SO 4, but this inexhaustible store is not much drawn upon; and the \"salt gardens\" on the coast of France lost their industrial importance as potash-producers since the deposi",
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"1222": {
"sentence.id": 1222,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The problem of its earlier history is so entangled with the similar questions raised by x that the two cannot well be discussed separately.",
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"1223": {
"sentence.id": 1223,
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"sentence.sentence": "This is a fairly strong case, but it falls short of demonstration because it cannot be shown that the MS. corrected by Pamphilus was still at Caesarea when it was used by x, and because it is not certain either that the chapter divisions in Acts were adde",
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"1224": {
"sentence.id": 1224,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Up to 1816-1817 these societies had printed altogether 436,000 copies of the Scriptures, and had received from the British and Foreign Bible Society gifts amounting to over X 62,000.",
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"1225": {
"sentence.id": 1225,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "That this unit is quite distinct from the Persian 86 grains is clear in the Egyptian weights, which maintain a wide gap between the two systems. Next, in Syria three inscribed weights of Antioch and Berytus (18) show a mina of about 16,400, or 200 x 82.",
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"1226": {
"sentence.id": 1226,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Pliny states the Egyptian talent at 80 librae = 396,000; evidently = the Abydus lion talent, which is divided by 100, and the mina is therefore 3960, or 50 x 79.2.",
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"1227": {
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"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "The largest weight is the \"wood\" talent of Syria (18) = 6 Roman talents, or 1,860,000, evidently 120 Antioch minae of 15,500 or 2 x 7750.",
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"1228": {
"sentence.id": 1228,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "So we have thus a weight of 207-191 in Egypt on marked weights, joining therefore completely with the Aeginetan unit in Egypt of 199 to 186, and coinage of 199, and strongly connected with Syria, where a double mina of Sidon (18) is 10,460 or 50 x 209.2.",
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"1229": {
"sentence.id": 1229,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "At Athens it was 2 x 4900, and on the average of all the Greek weights it is 2 x 4825, so that 4950 -- the libra -- is as close as we need expect.",
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"1230": {
"sentence.id": 1230,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "A beautiful set of multiples of the scripulum was found near Lyons (38), from 1 to 10 x 17.28 grains, showing a libra of 4976.",
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"1231": {
"sentence.id": 1231,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "Among them (29) the two contiguous groups can be discriminated by the 129 being multiplied by 30 and 60, while the 67 or 134 is differently x 25, 40, 50 and l00.",
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"1232": {
"sentence.id": 1232,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "This relation between x, a, rn, may be expressed also by the equation x= log m.",
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"1233": {
"sentence.id": 1233,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated.",
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"1234": {
"sentence.id": 1234,
"sentence.ts": "2022-12-13 05:45:53",
"sentence.sentence": "It follows very simply from the definition of a logarithm that logo b X logo a, = 1, logo m =log.",
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"1235": {
"sentence.id": 1235,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The mathematical function log x or log x is one of the small group of transcendental functions, consisting only of the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or sin-' x,&c., log x and e x which are universally treated in anal",
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"1236": {
"sentence.id": 1236,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The notation log x is generally employed in English and American works, but on the continent of Europe writers usually denote the function by lx or lg x.",
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"1237": {
"sentence.id": 1237,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The logarithmic function is most naturally introduced into analysis by the equation log x= x ?",
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"1238": {
"sentence.id": 1238,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "This equation defines log x for positive values of x; if o the formula ceases to have any meaning.",
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"1239": {
"sentence.id": 1239,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as algebraical or circular functions.",
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"1240": {
"sentence.id": 1240,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "A connexion with the circular functions, however, appears later when the definition of log x is extended to complex values of x.",
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"1241": {
"sentence.id": 1241,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The following fundamental properties of log x are readily deducible from the definition (i.) log xy= log x-Flog y.",
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"1242": {
"sentence.id": 1242,
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"sentence.sentence": "Either of these properties might be taken as itself the definition of log x.",
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"1243": {
"sentence.id": 1243,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "There is no series for log x proceeding either by ascending or descending powers of x, but there is an expansion for log (I +x), viz.",
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"1244": {
"sentence.id": 1244,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The function log x as x increases from o towards w steadily increases from - co towards +co.",
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"1245": {
"sentence.id": 1245,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "It has the important property that it tends to infinity with x, but more slowly than any power of x, i.e.",
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"1246": {
"sentence.id": 1246,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The exponential function possesses the properties (i.) exp (x+y) =exp x X exp y.",
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"1247": {
"sentence.id": 1247,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "It is customary, therefore, to denote the exponential function by e x, and the result ex = I +x+x2/2 !",
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"1248": {
"sentence.id": 1248,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The definitions of the logarithmic and exponential functions may be extended to complex values of x.",
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"1249": {
"sentence.id": 1249,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Thus even when the argument is real log x has an infinite number of values; for putting 71 =o and taking positive, in which case a = o, we obtain for log the infinite system of values log +2n7ri.",
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"1250": {
"sentence.id": 1250,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "It follows from this property of the function that we cannot have for log x a series which shall be convergent for all values of x, as is the case with sin x and cos x, for such a series could only represent a uniform function, and in fact the equation lo",
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"1251": {
"sentence.id": 1251,
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"sentence.sentence": "The exponential function, which may still be defined as the inverse of the logarithmic function, is, on the other hand, a uniform function of x, and its fundamental properties may be stated in the same form as for real values of x.",
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"sentence.sentence": "An alternative method of developing the theory of the exponential function is to start from the definition exp x = I +x+x2/2 !",
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"1253": {
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"sentence.sentence": "Logistic or Proportional Logarithms. - The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since",
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"sentence.sentence": "Corresponding to the argument log x it gives the values of log (I -Fx - 1) and log (1+x).",
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"sentence.id": 1255,
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"sentence.sentence": "He then by means of a simple proportion deduced that log (I 00000 00000 00000 I)=o 00000 00000 00000 0 434 2 944 81 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeate",
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"sentence.id": 1256,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Since loge(I +x) =x-2x 2 -3x 3 - 4x4+&c., we have, by changing the sign of x, log e (I - x) _ - x - zx 2 - 3x 3 - x 4 - &c.; whence g 1 +x to=2(x+ix'+1x5+&c.), e l - x and, therefore, replacing x by p +q, log e q =2 p +q +3 () 3T ?",
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"sentence.id": 1257,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1+.i\"x.",
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"1258": {
"sentence.id": 1258,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "X 67rapSos) was given by the ancients to an animal supposed to have been a cross between a lion (Lat.",
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"1259": {
"sentence.id": 1259,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Real information as to the nations of Mexico before Spanish 1 In this, as in all other Aztec names, the x (or j) represents the English sound sh; hence Mexitli and Mexico should be properly pronounced Meshitli, Meshico.",
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"1260": {
"sentence.id": 1260,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "But they do not appear to have ever been so pronounced by the Spaniards, who naturally gave to the x its ordinary Spanish sound of the German ch.",
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"1261": {
"sentence.id": 1261,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "In the centre, the oblong pyramid of rubble cased with hewn stone and cemented 375 X 300 ft.",
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"1262": {
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"sentence.sentence": "Seler, Verhandlungen des X VI.",
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"1263": {
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"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Each \" saying \" is introduced by the phrase Jesus says \" (XE'y€) and the collection is described in the introductory words of the 1903 series as Xoyyoc not as X&yca.",
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"1264": {
"sentence.id": 1264,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The sign of summation X is used in cases where there are several absorption bands, and consequently several similar terms on the right-hand side, each with a different value of A m.",
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"1265": {
"sentence.id": 1265,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "A portion of Pierce's correspondence has been published in the American Historical Review, x.",
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"1266": {
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"sentence.sentence": "The use of the words \" after being filled,\" in x.",
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"sentence.sentence": "But Paul, while he saw this much in it, saw much more; or he could not in the same epistle, x.",
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"1268": {
"sentence.id": 1268,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The two \"hospitia\" or \"guest-houses\" for the entertainment of strangers of different degrees (X, X2) comprise a large common chamber or refectory in the centre, surrounded by sleeping-apartments.",
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"1269": {
"sentence.id": 1269,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Between it and the transept we find the sacristy (X), and a small book-room (Y), armariolum, where the brothers deposited the volumes borrowed from the library.",
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"1270": {
"sentence.id": 1270,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Obviously these equations show that the curves intersect in four points, two of which lie on the intersection of the line, 2 (g - g')x +2 (f - f')y+c - c'=o, the radical axis, with the circles, and the other two where the lines x2+y2= (x+iy) (x - iy) =o (",
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"1271": {
"sentence.id": 1271,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The equation x 2 +y 2 =o denotes a pair of perpendicular imaginary lines; it follows, therefore, that circles always intersect in two imaginary points at infinity along these lines, and since the terms x 2 +y 2 occur in the equation of every circle, it is",
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"1272": {
"sentence.id": 1272,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Since the equation to a circle of zero radius is x 2 +y 2 =o, i.e.",
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"1273": {
"sentence.id": 1273,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The equations to such circles may be expressed in the form x 2 +y 2 = a 2, x 2 +y 2 = /3 2 .",
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"1274": {
"sentence.id": 1274,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "These equations show that the circles touch where they intersect the lines x 2 +y 2 = o, i.e.",
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"1275": {
"sentence.id": 1275,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle.",
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"1276": {
"sentence.id": 1276,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Analytically, the Cartesian equation to a coaxal system can be written in the form x 2 + y 2 + tax k 2 = o, where a varies from member to member, while k is a constant.",
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"1277": {
"sentence.id": 1277,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The radical axis is x = o, and it may be shown that the length of the tangent from a point (o, h) is h 2 k 2, i.e.",
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"1278": {
"sentence.id": 1278,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The value of the city's factory products increased from x$1,615,959 in 1900 to $3,146,890 in 1905, or 94-7%.",
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"1279": {
"sentence.id": 1279,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Internal pore leading from the non-glandular portion of the left nephridium to the external pore x.",
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"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The left inner gill-plate is also snipped to show the subjacent orifices of the left renal organ x, and of the genital gland (testis or ovary) y.",
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"1281": {
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"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "In the same figure the free part of the inner lamella of the inner gill-plate resting on the foot is marked z, whilst the attached parjt - the most anterior - has been snipped with scissors so as to show the genital and nephridial apertures x and y.",
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"1282": {
"sentence.id": 1282,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The readings of X which can be deduced from considering the agreements in B, C, D will be of higher antiquity and of greater external aut l ority than any of the readings in B, C, D taken singly.",
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"1283": {
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"sentence.sentence": "Glover, The Conflict of Religions tin the Early Roman Empire, chap. x.",
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"sentence.sentence": "Moreover, by adding (Politics, H 7, 1327 b 29-33) that the Greek race could govern the world by obtaining one constitution (was Tvy X b.vov 7roXtmeias), he indicated some leaning to a universal monarchy under such a king as Alexander.",
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"1285": {
"sentence.id": 1285,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Lastly, pleasure, after having been first defined (Book vii.) as an activity, is treated over again (Book x.) as an end beyond activity, with a warning against confusing activity and pleasure.",
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"1286": {
"sentence.id": 1286,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Secondly, the Eudemian Ethics, while not agreeing with Plato's Republic that the just can be happy by justice alone, does not assign to the external goods of good fortune (Eutu X ia) the prominence accorded to them in the Nicomachean Ethics as the necessa",
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"1287": {
"sentence.id": 1287,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "It divides (chap. 8) evidences (7r1aTEts) into two kinds (I) evidence from arguments, actions and men (ai j s v E air&v Tcev Xhywv cal Twv 7rpit aw cal TWV avOpcoirwv); (2) adventitious evidences (ai S' iIriOETOtTOLs X yo,u vots cal Tois rpm-To/lb/0a).",
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"1288": {
"sentence.id": 1288,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Aristotle was the founder of Logic; because, though others, and especially Plato, had made occasional remarks about reason (X yos), Aristotle was the first to conceive it as a definite subject of investigation.",
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"1289": {
"sentence.id": 1289,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "By Inserting, Therefore, Three Additional Months Instead Of Four In Every Period Of Eight Years, The Coincidence Between The Solar And Lunar Year Would Have Been Exactly Restored If The Latter Had Contained Only 354 Days, Inasmuch As The Period Contains 3",
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"1290": {
"sentence.id": 1290,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "On The Other Hand, The Exact Time Of A Synodic Revolution Of The Moon Is 29'530588 Days; 235 Lunations, Therefore, Contain 2 35 X 29 530588 = 6 939'6 8818 Days, Or 6 939 Days 16 Hours 31 Minutes, So That The Period Exceeds 235 Lunations By Only Seven And ",
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"1291": {
"sentence.id": 1291,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The Number Of Years In The Intercalary Period Being Four, And The Days Of The Week Being Seven, Their Product Is 4 X 7 = 28; Twenty Eight Years Is Therefore A Period Which Includes All The Possible Combinations Of The Days Of The Week With The Commencemen",
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"1292": {
"sentence.id": 1292,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "This Rule Is Conveniently Expressed By The Formula (X 9) R, In Which X Denotes The Date, And The Symbol R Denotes That The Remainder, Which Arises From The Division Of X 9 By 28, Is The Number Required.",
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"1293": {
"sentence.id": 1293,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The Symbol X, Employed In The Formula At The Top Of The Column, Denotes The Number Of Centuries, That Is, The Figures Remaining After The Last Two Have Been Struck Off.",
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"1294": {
"sentence.id": 1294,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "This gives 19 X354+6 X 30429 = 6 935 days, to be distributed among 235 lunar months.",
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"1295": {
"sentence.id": 1295,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The cycle of the sun brings back the days of the month to the same day of the week; the lunar cycle restores the new moons to the same day of the month; therefore 28 X 19 = 53 2 years, includes all the variations in respect of the new moons and the domini",
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"1296": {
"sentence.id": 1296,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "This number is called the Golden Number, either because will therefore be given by the formula (x 1 53) r, that is to say, add 3 totlzedate,divide the sum by 15, and the remainder is the year of the indiction.",
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"1297": {
"sentence.id": 1297,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The Julian period, proposed by the celebrated Joseph Scaliger as an universal measure of chronology, is formed by taking the continued product of the three cycles of the sun, of the moon, and of the indiction,and is consequently 28 X 19X I 5= 7980 years.",
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"1298": {
"sentence.id": 1298,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Let X, Y, And Z Be The Three Quotients Of The Divisions; The Number Sought Will Then Be Expressed By 28 X Io, By 19 Y 2, Or By 15 Z 4.",
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"1299": {
"sentence.id": 1299,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "To Solve The Equations 28 X Io =19 Y 2, Or Y =X 9198, Let M= 9 19 8, We Have Thenx=2 M 172 8 9 Let M9 8 =M'; Then M= 9 M' 8; Hence X=18 M' 16 M'=19 M' 16 (I).",
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"1300": {
"sentence.id": 1300,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Equating The Above Two Values Of X, We Have 15 N' 3 =19 M' 16; Whence N' =M' 4'N1 13.",
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"1301": {
"sentence.id": 1301,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "X, Period, 47 1 4, 47 1 5, 4716,...4713 X; From Which It Is Evident, That If We Take P To Represent The Year Of The Julian Period, And X The Corresponding Year Of The Christian Era, We Shall Have P= 47 1 3 X, And X=P 4713.",
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"1302": {
"sentence.id": 1302,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "In The Year Preceding The First Of The Era, The Dominical Letter Was C; For That Year, Therefore, We Have L =3; Consequently For Any Succeeding Year X, L =7M 3 X, The Years Being All Supposed To Consist Of 365 Days.",
"sentence.uniq": "118953ad887e24ac03150da468a4bd7c"
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"1303": {
"sentence.id": 1303,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The Above Expression Must Therefore Be Diminished By The Number Of Units In 4, Or By () W (This Notation Being Used To Denote The Quotient, In A Whole Number, That Arises From Dividing X By 4).",
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"1304": {
"sentence.id": 1304,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Hence In The Julian Calendar The Dominical Letter Is Given By The Equation L= 7M 3 X () W This Equation Gives The Dominical Letter Of Any Year From The Commencement Of The Era To The Reformation.",
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"1305": {
"sentence.id": 1305,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "After X Centuries, Therefore, It Must Be Diminished By X (4).",
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"1306": {
"sentence.id": 1306,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "This Gives Eight To Be Added In A Period Of Twenty Five Centuries, And 85 In X Centuries.",
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"1307": {
"sentence.id": 1307,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "P. Pfeiffer (Berichte, 1904, 37, p. 4 2 55) has shown that chromium salts of the type [Cr{C2H4(NH2)2}2X2]X exist in two stereo-isomeric forms, namely, the cisand transforms, the dithiocyan-diethylenediamine-chromium salts being the transsalts.",
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"1308": {
"sentence.id": 1308,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Let somehow or other retardations be introduced so that the optical length of the successive parts increases by the same quantity nX, n being some number and X the wave-length.",
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"1309": {
"sentence.id": 1309,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "B, a specimen undergoing fission (X 20).",
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"1310": {
"sentence.id": 1310,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Grignard (Comptes rendus, 1900 et seq.) observed that magnesium and alkyl or aryl halides combined together in presence of anhydrous ether at ordinary R temperatures (with the appearance of brisk boiling) to form compounds of the type RMgX(R = an alkyl or",
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"1311": {
"sentence.id": 1311,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The Libel of English Policie, a poem of the first half of the 15th century, says with reference to Iceland (chap. x.) \"Out of Bristowe, and costes many one, Men haue practised by nedle and by stone Thider wardes within a litle while.\"",
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"1312": {
"sentence.id": 1312,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "He took as his secretary of state Cardinal Raphael p us x Merry del Val, a Spaniard of English birth and educa tion, well versed in diplomacy, but of well-known ultramontane tendencies.",
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"1313": {
"sentence.id": 1313,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The personality of Lazarus in John's account, his relation to Martha and Mary, and the possibility that John reconstructed the story by the aid of inferences from the story of the supper in Luke x.",
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"1314": {
"sentence.id": 1314,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "The warp is commonly from 36 8 to 44 8, the weft from 36 8 to 54 8, and the threads from 13 X 13 to 20X 20 to the 4 in.",
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"1315": {
"sentence.id": 1315,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "Arabic and other writers are given in Ritter, Erdkunde x.",
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"1316": {
"sentence.id": 1316,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "It is practically only bulls of canonization which are signed by the pope and all the cardinals present in Rome; the signature of the pope is then \"(Pius) Episcopus Ecclesiae catholicae,\" while his ordinary signature bears only his name and number, \"Pius ",
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"1317": {
"sentence.id": 1317,
"sentence.ts": "2022-12-13 05:45:54",
"sentence.sentence": "But, if no solid be present initially, or if the cooling be rapid, the liquid of composition x becomes supersaturated and may cool till the supersaturation curve is reached at b, and a cloud of A crystals comes down.",
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"sentence.sentence": "Xander took my door off a week ago.",
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"sentence.sentence": "And if it is Xander, maybe the Others want something from Xander that Xander himself doesn't know he can do for them.",
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"sentence.sentence": "She'd been planting hints for days that Xander couldn't be trusted.",
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"sentence.sentence": "She'd never met anything like Xander, who was not deterred by even a god's powers.",
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"sentence.sentence": "Maybe. But between Xander and me, he's got more power than you and I combined.",
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"1426": {
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"sentence.sentence": "Jonny didn't seem to know her fate, which meant Xander hadn't told her secret, that she was intended to mate with the Grey God.",
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"1427": {
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"sentence.sentence": "They were passive aggressive with her, hazing her when the Black God and Xander weren't around.",
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"1428": {
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"sentence.sentence": "Xander was in the gym whenever she was, no matter what time of day or night she went.",
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"sentence.sentence": "I'm not here to entertain you, Xander.",
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"1430": {
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"sentence.sentence": "Jenn joined Xander in the ring.",
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"sentence.sentence": "Xander deflected and attacked.",
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"1432": {
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"sentence.sentence": "Xander's elbow smashed into her face.",
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"1433": {
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"sentence.sentence": "When Xander didn't look away, she began to suspect she'd found a weak point.",
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"1434": {
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"sentence.sentence": "She'd stopped trying to figure out why Xander tormented her every day.",
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"1435": {
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"sentence.sentence": "She rose and gasped, not sensing the silent Xander, who stood in the doorway to the living room.",
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"1436": {
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"sentence.sentence": "I don't care, Xander.",
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"1437": {
"sentence.id": 1437,
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"sentence.sentence": "As she hacked at a young tree, she thought of Xander's words.",
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"sentence.sentence": "For an Original Being, Xander has no discretion.",
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"1439": {
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"sentence.sentence": "Dammit, Xander, Sofi muttered.",
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"1440": {
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"sentence.sentence": "A small part of her suspected Damian knew what Xander knew.",
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"1441": {
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"sentence.sentence": "Xander can't even stop them!",
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"1442": {
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"sentence.sentence": "If Jonny got you, then what did Damian … Xander.",
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"1443": {
"sentence.id": 1443,
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"sentence.sentence": "The vamps only try shit when Jonny and Xander aren't around.",
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"1444": {
"sentence.id": 1444,
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"sentence.sentence": "If it's not them, it was Xander kicking my ass in the gym.",
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"1445": {
"sentence.id": 1445,
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"sentence.sentence": "He had no sense of his own mortality and fighting skill that rivaled Xander's.",
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"1446": {
"sentence.id": 1446,
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"sentence.sentence": "Xander had promised Jule to behave, an oath Jule said he couldn't break, since it was Original-to-Original.",
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"1447": {
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"sentence.sentence": "I'll keep your secrets, Xander.",
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"1448": {
"sentence.id": 1448,
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"sentence.sentence": "I've Seen all your possible fates, Xander.",
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"1449": {
"sentence.id": 1449,
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"sentence.sentence": "Sofi doubted Pierre would be living if Xander hadn't made a promise.",
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"1450": {
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"sentence.sentence": "Xander would never help another without some sort of leverage.",
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"1451": {
"sentence.id": 1451,
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"sentence.sentence": "She hesitated, aware she never should've told Damian, let alone allowed Xander to pry the answer out of her when they'd met.",
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"1452": {
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"sentence.sentence": "Xander had gone so far to tell Jenn.",
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"1453": {
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"sentence.sentence": "Sofi, any reason why I shouldn't take Xander with me?",
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"1454": {
"sentence.id": 1454,
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"sentence.sentence": "Basically, he trusts me fifty percent of the time, Xander fifty percent of the time, and no one else.",
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"1455": {
"sentence.id": 1455,
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"sentence.sentence": "I take it you and Xander don't necessarily see eye-to-eye.",
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"1456": {
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"sentence.sentence": "He'd taken Xander with him twice.",
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"1457": {
"sentence.id": 1457,
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"sentence.sentence": "She wasn't going down the path Xander said was hers.",
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"1458": {
"sentence.id": 1458,
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"sentence.sentence": "Where Xander used brute force to subdue an opponent, Darian was wily.",
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"1459": {
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"sentence.sentence": "Jenn adjusted her stance, keeping more of a distance from him than she would Xander to give her more time to react.",
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"1460": {
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"sentence.sentence": "Xander taught me how to use my body.",
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"sentence.sentence": "Xander didn't pull punches, Darian.",
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"sentence.sentence": "Not only was an Other there, but so was Damian, the messenger he'd sent, Jonny, Jenn, and Xander.",
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"1463": {
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"sentence.sentence": "Xander shifted from across the room, his red eyes punctuating the darkness.",
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"1464": {
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"sentence.sentence": "Pretty sure that's a rule, too, right, Xander?",
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"1465": {
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"sentence.sentence": "Jonny waited for Xander to go before he, too, disappeared.",
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"sentence.sentence": "Xander told me everything.",
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"1467": {
"sentence.id": 1467,
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"sentence.sentence": "Now, read my entry from the day I met Xander, four weeks ago today.",
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"1468": {
"sentence.id": 1468,
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"sentence.sentence": "This time, Xander says Darian's fate is not the same as his predecessor.",
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"1469": {
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"sentence.sentence": "Xander says they can't kill him, but maybe they can keep him from destroying our world.",
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"1470": {
"sentence.id": 1470,
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"sentence.sentence": "They didn't talk down to me like they did when Xander was at our meetings.",
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"1471": {
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"sentence.sentence": "They used mind manipulation to control you, which wasn't possible with Xander there to protect you.",
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"sentence.id": 1472,
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"sentence.sentence": "Xander can still do it.",
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"sentence.sentence": "I'd rather face ten guardsmen than one Xander.",
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"1474": {
"sentence.id": 1474,
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"sentence.sentence": "She couldn't sense them as she did when her magic was free, but she could dance the way Darian and Xander taught her.",
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"sentence.sentence": "You're talking about Xander, right?",
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"sentence.sentence": "Jenn asked, startled as the Oracle pushed her aside to stand before Xander.",
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"sentence.sentence": "Xander's bite hadn't been gentle.",
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"sentence.sentence": "The woman radiated cold power, her eyes brighter than even Xander's.",
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"sentence.sentence": "Jenn touched Xander's arm.",
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"sentence.id": 1481,
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"sentence.sentence": "After the exchange in the immortal world, she knew Sofi was on no good terms with Xander.",
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"sentence.sentence": "Xander snatched her arm, and she closed her eyes.",
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"1483": {
"sentence.id": 1483,
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"sentence.sentence": "His magic filled the air, shoving between Bianca and Xander and pushing her away.",
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"sentence.sentence": "I felt it this time, Xander.",
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"1485": {
"sentence.id": 1485,
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"sentence.sentence": "Whatever the exchange between him and Xander meant, he'd showed how strong he'd grown.",
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"sentence.id": 1486,
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"1487": {
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"sentence.sentence": "Darian looked at Xander, unable to determine what game the Original Being played.",
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"sentence.id": 1488,
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"sentence.sentence": "Darian strode forward, Xander forgotten as he focused on the two women.",
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"sentence.sentence": "He didn't look back but tagged Xander on the arm as he passed.",
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"1490": {
"sentence.id": 1490,
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"sentence.sentence": "Xander watched her approach, seemingly at ease while the world went to shit around him.",
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"sentence.sentence": "Xander's red eyes went to the unconscious god.",
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"1492": {
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"sentence.sentence": "Xander bent and hefted Darian over his shoulder.",
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"1493": {
"sentence.id": 1493,
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"sentence.sentence": "She needed to know what happened to Darian, if Xander followed through and delivered the Grey God home.",
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"1494": {
"sentence.id": 1494,
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"sentence.sentence": "Bianca and Sofi shared a look before Sofi said grudgingly, \"Xander brought you.\"",
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"1495": {
"sentence.id": 1495,
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"sentence.sentence": "Masked and hooded to hide his deformity, Xander relied on his special senses, the ones that no one else possessed.",
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"1496": {
"sentence.id": 1496,
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"sentence.sentence": "Xander was forced to learn to use his special skills to steal from the market's patrons rather than beg with the rest of the kids.",
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"1497": {
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"sentence.sentence": "Xander yanked the door open and froze.",
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"1498": {
"sentence.id": 1498,
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"sentence.sentence": "Her kidskin boots alone were worth more than everything Xander had ever stolen combined.",
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"1499": {
"sentence.id": 1499,
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"sentence.sentence": "They were pale green with silver rings that seemed to liquefy and swirl as Xander watched.",
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"1500": {
"sentence.id": 1500,
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"sentence.sentence": "Xander's surprise turned to concern for his mother.",
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"1501": {
"sentence.id": 1501,
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"sentence.sentence": "Xander crossed to his mother and knelt, wary of the stranger.",
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"1502": {
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"sentence.sentence": "Xander's eyes were glued to his mother.",
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"1503": {
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"sentence.sentence": "Xander held his mother's hand and lifted his gaze.",
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"1504": {
"sentence.id": 1504,
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"sentence.sentence": "It was not a warm smile, like Xander's mother gave him, but a kind of smile that left Xander scared, without knowing why.",
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},
"1505": {
"sentence.id": 1505,
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"sentence.sentence": "Xander barely resisted the urge to touch the finely woven garment with a fur lining that was certain to be the softest thing in the world.",
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},
"1506": {
"sentence.id": 1506,
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"sentence.sentence": "Xander was instantly fascinated by the sensation of downy fur and cotton spun so finely, it was like silk.",
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"sentence.sentence": "The stranger's face softened with the warmth of pity, a sight Xander was accustomed to.",
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"sentence.sentence": "He rarely thought of the man he didn't remember, but since the stranger's visit, Xander wasn't able to get his father out of his mind.",
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"sentence.sentence": "From this point, Xander was able to see most of the city, including the white dome of the palace at its center that marked the home of one of the three Gods that ruled the immortal realm.",
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"sentence.sentence": "Will you do something for me, Xander?",
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"sentence.sentence": "If Xander hadn't erased the events calendar, she'd know where he'd be so she could search his house for the necklace.",
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"sentence.sentence": "After all, the thought of kicking the annoying woman, Toni, out of Xander's bed and apartment sounded really appealing.",
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"sentence.sentence": "Xander spent the afternoon with Toni and then made the journey he didn't want to.",
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"sentence.sentence": "Xander's ability was far weaker than hers, but he was able to see certain parts of another's path when in their minds.",
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"sentence.sentence": "Eden grinned, appearing even younger than Xander remembered.",
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"sentence.sentence": "Troubled, Xander didn't bother taking his snack to the table but wolfed it down over the sink, too troubled to care about the mess he made.",
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"sentence.sentence": "What did Xander do to these women?",
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