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formula

formula Sentence Examples

  • Every morning before I went to school I had chores to do, which began with mixing up the formula and feeding the calves.

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  • The students could not figure out the math problem until their teacher provided them with a helpful formula.

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  • The baby was unable to breastfeed, so her mother had to use formula instead.

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  • In point of fact, bourgeois democracy is the political formula for free trade, nothing more.

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  • Aiden's mother sent a big tub of formula to daycare so he would not get hungry.

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  • The chemical formula shows all the elements and compounds involved in the reaction.

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  • There may be an advantage to the divorce action being filed in a certain jurisdiction; for example a couple may not want to file in a community property state if they have agreed on a different formula for dividing their marital property.

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  • In Switzerland secret remedies cannot be advertised without submitting the formula and a sample of the remedy to the board of health.

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  • Although an easy way to introduce yourself to the wonderful world of hair color, a semi-permanent hair coloring formula is still a commitment and should not be done without thought to the long term effects that it will have on your hair.

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  • I bought some lamb milk formula.

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  • Bredig point to the salt possessing the double formula.

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  • - The formula for the rth term is easily obtained.

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  • Other objections to Ladenburg's formula resulted from A.

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  • The exact formula for the papal infallibility is given by the Vatican Council in the following terms (Constit.

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  • The pair 6232 and 6368 are amicable, but they cannot be derived from this formula.

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  • C. Palazzo (1907) support this formula, finding that methyl nitrolic acid, NO 2 CH: N-OH, yielded under certain conditions fulminic acid, and vice versa (Palazzo, 1907).

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  • Spencer's grandiose cosmic formula in terms of mechanism.

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  • Aspinall's results are expressed by the formula S 5+ 50.8 12 r„ =2 -}-0 0278,L where r ro is the resistance in pounds per ton, S is the speed in miles per hour, and L is the length of the train in feet measured over the carriage bodies.

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  • 17, four values of L being taken for formula (12) corresponding to trains of 5, 10, 15 and 20 bogie carriages.

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  • This formula is valid between speeds of 37 and 77 m.

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  • Barbier's formula is plotted in fig.

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  • which the area represents from the following formula, which is expressed in terms of the absolute temperature T, of the steam at the steam-pipe, and the temperature T2= 461°H-212° =673° absolute corresponding to the back pressure: - Maximu per pound oflsteaelwork =U=(T - T) (i -f -Lr--.1) - T2loge.Tr--2.

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  • That the substance of the Physiologus was borrowed from commentaries on Scripture 4 is confirmed by many of the sections opening with a text, followed up by some such formula as "but the Physiologus says."

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  • Like Berthelot, he writes the chemical equation of the reaction, but in addition he considers the chemical formula of each substance to express not only its material composition, but also the (unknown) value of its intrinsic energy.

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  • Cane-sugar has the formula C12H22011.

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  • Berthelot, and many other chemists, from whose researches it results that glycerin is a trihydric alcohol indicated by the formula C 3 H 5 (OH) 3j the natural fats and oils, and the glycerides generally, being substances of the nature of compound esters formed from glycerin by the replacement of the hydrogen of the OH groups by the radicals of certain acids, called for that reason "fatty acids."

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  • It is an oxide of iron having the formula Fe 3 O 4, corresponding with 72.4% of metal, whence its great value as an ore.

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  • Closely related to magnetite is the rare volcanic mineral from Vesuvius, called magnoferrite, or magnesioferrite, with the formula MgFe 2 O 4; and with this may be mentioned a mineral from Jakobsberg, in Vermland, Sweden, called jakobsite, containing MnFe204.

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  • In systematic chemistry, sodium hyposulphite is a salt of hyposulphurous acid, to which Schutzenberger gave the formula H 2 S0 2, but which Bernthsen showed to be H 2 S 2 0 4.

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  • But he was no merely destructive critic. He was determined to find a solid foundation for both morality and law, and to raise upon it an edifice, no stone of which should be laid except in accordance with the deductions of the severest logic. This foundation is "the greatest happiness of the greatest number," a formula adopted from Priestly or perhaps first from Beccaria.

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  • " Thomson and Tait," as it is familiarly called ("T and T" was the authors' own formula), was planned soon after Lord Kelvin became acquainted with Tait, on the latter's appointment to his professorship in Edinburgh, and it was intended to be an all-comprehensive treatise on physical science, the foundations being laid in kinematics and dynamics, and the structure completed with the properties of matter, heat, light, electricity and magnetism.

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  • His general formula for getting at the number of units in any sensation is S = C log R, where s stands for the sensation, R for the stimulus numerically estimated, and c for a constant that must be separately determined by experiment in each particular order of sensibility.

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  • Formula 3: I: 3.

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  • Formula I: I: 1.

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  • (Formula, x.l.x.) radula has a number of uniform teeth on each side of the median tooth in each transverse row.

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  • The assembly protested, but was appeased by the empty formula, "This is your doge an it please you."

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  • 2 The words "substantially converted" appear in the formula which Berengarius was compelled to sign in 1079.

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  • BOOK OF CONCORD (Liber Concordiae), the collective documents of the Lutheran confession, consisting of the Confessio Augustana, the Apologia Confessionis Augustanae, the Articula Smalcaldici, the Catechismi Major et Minor and the Formula Concordiae.

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  • This last was a formula issued on the 25th of June 1580 (the jubilee of the Augsburg Confession) by the Lutheran Church in an attempt to heal the breach which, since the death of Luther, had been widening between the extreme Lutherans and the Crypto-Calvinists.

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  • Previous attempts at concord had been made at the request of different rulers, especially by Jacob Andrea with his Swabian Concordia in 1573, and Abel Scherdinger with the Maulbronn Formula in 1575.

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  • In spite of this very limited reception the Formula Concordiae has always been reckoned with the five other documents as of confessional authority.

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  • Old Roman formula of prayer mention a Hora Quirini, his female cult associate, afterwards identified with Hersilia, the wife of Romulus.

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  • Thus the pseudo-Democritus, who was reputed the author of the Physica et Mystica, which itself concludes each of its receipts with a magical formula, was believed to have travelled in Chaldaea, and to have had as his master Ostanes l the Mede, a name mentioned several times in the Leiden papyrus, and often by early Christian writers such as Tertullian, St Cyprian and St Augustine.

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  • The basic oxides must have the general formula MO.

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  • The molecular formula of a compound, however, is always a simple multiple of the empirical formula, if not identical with it; thus, the empirical formula of acetic acid is CH 2 O, and its molecular formula is C2H402, or twiceTCH 2 O.

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  • For instance, sulphuric acid is usually represented by the formula S0 2 (OH) 2, which indicates that it may be regarded as a compound of the group SO 2 with twice the group OH.

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  • Gomberg's triphenyl-methyl play no part in what follows), it is readily seen that the simplest hydrocarbon has the formula CH 4, named methane, in which the hydrogen atoms are of equal value, and which may be pictured as placed at the vertices of a tetrahedron, the carbon atom occupying the centre.

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  • This tetrahedral configuration is based on the existence of only one methylene dichloride, two being necessary if the carbon valencies were directed from the centre of a plane square to its corners, and on the existence of two optical isomers of the formula C. A.

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  • Henry started with methyl iodide, the formula of which we write in the form CI a H b H c H d.

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  • It will be noticed that compounds containing two double linkages will have the same general formula as the acetylene series; such compounds are known as the " diolefines."

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  • they possess the same formula, but differ in constitution.

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  • The values in other cases are calculable from the formula RI 09° 28' - a), where a is the internal angle of the regular polygon contained by sides equal in number to the number of the carbon atoms composing the ring.

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  • This formula, notwithstanding many attempts at both disproving and modifying it, has well stood the test of time; the subject has been the basis of constant discussion, many variations have been proposed, but the original conception of Kekule remains quite as convenient as any of the newer forms, especially when considering the syntheses and decompositions of the benzene complex.

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  • By fusing two nuclei we obtain the formula of naphthalene, C 1 oH 8; by fusing three, the hydrocarbons anthracene and phenanthrene, C14H10; by fusing four, chrysene, C18H12, and possibly pyrene, C16H1n; by fusing five, picene, C22 H 14.

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  • Although Kekule founded his famous benzene formula in 1865 on the assumptions that the six hydrogen atoms in benzene are equivalent and that the molecule is symmetrical, i.e.

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  • If we accept Kekule's formula for the benzene nucleus, then we may expect the double linkages to be opened up partially, either by oxidation or reduction, with the formation of di-, tetra-, or hexa-hydro derivatives, or entirely, with the production of open chain compounds.

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  • Claus (Theoretische Betrachtungen and deren Anwendung zur Systematik der organischen Chemie) proposed his diagonal formula (III), and two years later, A.

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  • Ladenburg (Ber., 2, p. 140) devised his prism formula (IV), the six carbon atoms being placed at the six corners of a right equilateral triangular prism, with its plane projections (V, VI).

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  • CH CH HC, N CH HC CH HC HC HC Hcch Hc Ch Ch Kekule t i Dewar Ladenburg One of the earliest and strongest objections urged against Kekule's formula was that it demanded two isomeric ortho-di-substitution derivatives; for if we number the carbon atoms in cyclical order from i to 6, then the derivatives 1.2 and 1.6 should be different.'

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  • Therefore, according to Kekule, the double linkages are in a state of continual oscillation, and if his dynamical notion of valency, or a similar hypothesis, be correct, then the difference between the 1.2 and 1.6 di-derivatives rests on the insufficiency of his formula, which represents the configuration during one set of oscillations only.

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  • Ladenburg, who devoted much attention to the study of the substitution products of benzene, and 's to the support of his own formula.

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  • The prism formula also received support from the following data: protocatechuic acid when oxidized by nitrous acid gives carboxytartronic acid, which, on account of its ready decomposition into carbon dioxide and tartronic acid, was considered to be HO C(COOH) 3.

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  • This implied that in the benzene complex there was at least one carbon atom linked to three others, thus rendering Kekule's formula impossible and Ladenburg's and Claus' possible.

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  • The formation of this substance readily follows from Kekule's formula, while considerable difficulties are met with when one attempts an explanation based on Ladenburg's representation.

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  • Strecker to be trichloracetoacrylic acid, was more favourably explained by his formula than by Ladenburg's.

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  • that 1.2 and 1.6 should be ortho-positions, 1.3 and 1.5 meta-, and 1.4 para-, and following out the transformation on the Ladenburg formula, then an ortho-dioxyterephthalic acid (IV) should result, a fact denied by experience, and inexplicable unless we assume a wandering of atoms. Kekule's formula (III), on the other hand, is in full agreement (Baeyer).

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  • From these results Baeyer concluded that Claus' formula with three para-linkings cannot possibly be correct, for the Q2.5 dihydroterephthalic acid undoubtedly has two ethylene linkages, since it readily takes up two or four atoms of bromine, and is oxidized in warm aqueous solution by alkaline potassium permanganate.

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  • The stronger argument against the ethylenoid linkages demanded by Kekule's formula is provided by the remarkable stability towards oxidizing and reducing agents which characterizes all benzenoid compounds.

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  • In this formula, the so-called " centric formula," the assumption made is that the fourth valencies are simply directed towards the centre of the ring; nothing further is said about the fourth valencies except that they exert a pressure towards the centre.

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  • Claus maintained that Baeyer's view was identical with his own, for as in Baeyer's formula, the fourth valencies have a different function from the peripheral valencies, being united at the centre in a form of potential union.

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  • It is difficult to determine which configuration most accurately explains the observed facts; Kekule's formula undoubtedly explains the synthetical production of benzenoid compounds most satisfactorily, and W.

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  • Marckwald (Ann., 93, 74, 331; 1894, 79, 14) has supported this formula from considerations based on the syntheses of the quinoline ring.

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  • Bamberger (a strong supporter of the centric formula), have shown that the nature of the substituent groups influences the distribution of the fourth valencies; therefore it may be con-?

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  • A similar contradiction apparently exists with regard to the specific volume, for while benzene has a specific volume correspinding to Claus' formula, toluene, or methylbenzene, rather points to Kekule's.

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  • Stohmann of Leipzig; and the new data and the conclusions to be drawn from them formed the subject of much discussion, Briihl endeavouring to show how they supported Kekule's formula, while Thomsen maintained that they demanded the benzene union to have a different heat of combustion from the acetylene union.

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  • It was found that the results were capable of expression by the empirical relation CaH2b= 104.3b+49'09m+105.47n, where C a H 2b denotes the formula of the hydrocarbon, m the number of single carbon linkings and n the number of double linkings, m and n being calculated on the Kekule formulae.

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  • Two acids corresponding to the formula of Kekule and Claus are triphenyl acrylic acid, (C6H5)2C: C([[Cooh) C 6 H]],, and triphenyl acetic acid, (C,H,),C 000H.

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  • Experiments showed that the second acid was much more difficult to esterify than the first, pointing to the conclusion that Claus' formula for benzene was more probable than Kekule's.

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  • Ladenburg's prism formula would give two enantiomorphic ortho-di-substitution derivatives; while forms in which the hydrogen atoms are placed at the corners of a regular octahedron would yield enantiomorphic tri-substitution derivatives.

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  • The octahedral formula discussed by Julius Thomsen (Ber., 1886, 19, p. 2 944) consists of the six carbon atoms placed at the corners of a regular octahedron, and connected together by the full lines as shown in (I); a plane projection gives a hexagon with diagonals (II).

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  • Arrangements connected with Claus' formula are obtained by placing six tetrahedra on the six triangles formed by the diagonals of a plane hexagon.

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  • Therefore formula (2), being unsymmetrical, is impossible.

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  • The third formula is based on Dewar's benzene formula, which we have seen to be incorrect.

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  • Formula (4) is symmetrical and based on Kekule's formula: it is in full accord with the syntheses and decompositions of the naphthalene nucleus and the number of isomers found.

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  • Bamberger opposed Claus' formula on the following grounds: - The molecule of naphthalene is symmetrical, since 2.7 dioxynaphthalene is readily esterified by methyl iodide and sulphuric acid to a dimethyl ether; and no more than two mono-substitution derivatives are known.

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  • It remains, therefore, to consider Erlenmeyer's formula and those derived from the centric hypothesis.

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  • The centric formula proposed by Bamberger represents naphthalene as formed by the fusion of two benzene rings, this indicates that it is a monocyclic composed of ten atoms of carbon.

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  • The formula has the advantage that it may be constructed from tetrahedral models of the carbon atom; but it involves the assumption that the molecule has within it a mechanism, equivalent in a measure to a system of railway points, which can readily close up and pass into that characteristic of benzene.

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  • These isomeric hydrocarbons, of the formula C14H10, are to be regarded as formed by the fusion of three.

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  • 456) has given the formula Cp=6.5--aT, where a is a constant depending on the complexity of the molecule, as an expression for the molecular heat at constant pressure at any temperature T (reckoned on the absolute scale).

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  • To take an example: 38 parts of indium combine with 35.4 parts of chlorine; hence, if the formula of the chloride be InCI, InC1 2 or InC1 3, indium has the atomic weights 38, 76 or 114.

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  • Consider now the combustion of a hydrocarbon of the general formula CH 2m.

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  • methylene oxide if we assign to it the formula H 2 C O CH 2, but if the formula H 2 C O CH 2 (which assumes the presence of two free valencies) be accepted, the calculated and observed heats of formation are in agreement.

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  • P. Dale; the more simple formula (n - i)/d, which remained constant for gases and vapours, but exhibited slight discrepancies when liquids were examined over a wide range of temperature, being adopted.

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  • The subject was next taken up by Hans Landolt, who, from an immense number of observations, supported in a general way the formula of Gladstone and Dale.

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  • To determine N recourse must be made to Cauchy's formula of dispersion (q.v.), n =A+B/X2+C/A4+...

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  • It is found experimentally that the Lorenz and Lorentz function holds fairly well, and better than the Gladstone and Dale formula.

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  • This is shown by the following observations of Riihlmann on water, the light used being the D line of the spectrum: Eykmann's observations also support the approximate constancy of the Lorenz-Lorentz formula over wide temperature differences, but in some cases the deviation exceeds the errors of observation.

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  • The values are for the Ha line: The empirical formula (n2-I)/(n2-1-o 4)d apparently gives more constant values with change of temperature than the LorenzLorentz form.

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  • The superiority of the Lorenz-Lorentz formula over the Gladstone and Dale formula for changes of state is shown by the following observations of Briihl (Zeit.

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  • In the free state these substances are colourless, and were assumed to have the formula shown in i.

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  • The ordinary structural formula of potassium sulphate is K - O - S - O - K.

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  • He first appears (c. 357) as a supporter of Acacius, bishop of Caesarea, the leader of that party in the episcopate which supported the Homoean formula by which the emperor Constantius sought to effect a compromise between the Homoeusians and the Homousians.

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  • According to Socrates he attended the synod of Seleucia in the autumn of 359, and then subscribed the Acacian formula.

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  • The Meletian schism was complicated, moreover, by the presence in the city of another anti-Arian sect, stricter adherents of the Homousian formula, maintaining the tradition of the deposed bishop Eustathius and governed at this time by the presbyter Paulinus.

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  • A common formula has i 1 parts of the acid, 3 of extract of Indian hemp, and 86 of collodion.

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  • The raw materials used in the manufacture are: (I) iron-free kaolin, or some other kind of pure clay, which should contain its silica and alumina as nearly as possible in the proportion of 2SiO 2: Al203 demanded by the formula assigned to ideal kaolin (a deficit of silica, however, it appears can be made up for by addition of the calculated weight of finely divided silica); (2) anhydrous sulphate of soda; (3) anhydrous carbonate of soda; (4) sulphur (in the state of powder); and (5) powdered charcoal or relatively ash-free coal, or colophony in lumps.

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  • In all other cases recourse must be had to a map, a globe or mathematical formula.

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  • The formula is Fe n, Sn+, where n may vary from 5 to 16; usually it is Fe, S8 or Fe l s S12, the latter being also the composition of the artificially prepared compound.

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  • In the Titanotheroidea the dentition may be expressed by the formula i: oru, ci, p T 4,, my.

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  • Finally, we have the family Rhinocerotidae, which includes the existing representatives of the group. In this family the dentition has undergone considerable reduction, and may be represented inclusive of all the variations, by the formula i a or a m a The first upper incisor, whenpresent, has an 430r2; PP antero-posteriorly elongated crown, but the second is small; when fully developed, the lower canine is a large forwardly directed tusk-like tooth with sharp cutting-edges, and biting against the first upper incisor.

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  • Thus, as Harnack points out, "there is no trace of a theological difference between Severus and Leontius," only a difference of terminology and of degree of willingness to assent to the formula of Chalcedon.

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  • A third variety of monophysitism was that known as Theopaschitism, a name given to those who accepted the formula that in the death of Christ "God had suffered and been crucified."

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  • In general it may be said, then, that whereas exorcism is practised in order to cast out devils already in possession, benediction is the formula by which they are prevented from entering in.

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  • 18.14%, S=0.96%, O = 22.66%, which points to the formula C720H1134N218S50241, corresponding to the molecular weight 16,954.

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  • By a dilute acid haemoglobin is decomposed into globin, and " haematin," a ferri-pyrrol derivative of the probable formula C34H34N4FeOs; under certain conditions the iron-free " haematoporphyrin " is obtained.

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  • The salt must therefore be derived from an acid, chloroplatinic acid, H 2 PtC1 6, and have the formula Na 2 PtC1 6, the ions being Na and PtCls", for if it were a double salt it would decompose as a mixture of sodium chloride and platinum chloride and both metals would go to the cathode.

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  • If some of the anions, instead of being simple iodine ions represented chemically by the symbol I, are complex structures formed by the union of iodine with unaltered cadmium iodide - structures represented by some such chemical formula as I(CdI 2), the concentration of the solution round the anode would be increased by the passage of an electric current, and the phenomena observed would be explained.

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  • Van 't Hoff's formula is equivalent to taking the frequency of dissociation as proportional to the square of the concentration of the molecules, and the frequency of recombination as proportional to the cube of the concentration of the ions.

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  • The forces between the ions of a strongly dissociated solution will thus be considerable at a dilution which makes forces between undissociated molecules quite insensible, and at the concentrations necessary to test Ostwald's formula an electrolyte will be far from dilute in the thermodynamic sense of the term, which implies no appreciable intermolecular or interionic forces.

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  • yXiKis, sweet), a carbohydrate of the formula C 6 H 12 0 6; it may be regarded as the aldehyde of sorbite.

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  • 0.85 3.20 i 6.9 0.3 7.7 84'3 11.8080 8 2.42 0.20.3 the formula C5H8.

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  • The properties of caoutchouc clearly show, however, that its actual molecular structure is considerably more complex than is represented by the empirical formula, and that it is to be regarded as the polymer of a terpene or similar hydrocarbon and composed of a cluster of at least ten or twenty molecules of the formula C5H8.

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  • From this formula we obtain by elementary algebra 1) !

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  • This proves, also, that in any formula connecting a 1, a 2, a 3, ...

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  • The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3!

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  • 7r corresponding to Thomas Waring's formula for the single system.

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  • The analogous formula appertaining to n systems of quantities which Vienna Transactions, t.

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  • We can verify the relations s 30 -a310 -3a 20 a 10 + a30, S 21 - 02100 01 -a 2C a 01 -0 11 0 10 021 The formula actually gives the expression of q) by means of separations of (10P01'), which is one of the partitions of (pq).

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  • = exp {(siox+Solt') - s20 x 2+ 2siixy+S02y2)+���}, and thence derive the formula?

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  • The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.

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  • Calling the discriminate D, the solution of the quadratic as =o is given by the formula a: = o (a0+a12_x2 (a0x+aix2 If the form a 2 be written as the product of its linear factors p.a., the discriminant takes the form -2(pq) 2.

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  • For it is easy to establish] the formula (yx) 2 0 4 = 2f.4-2(f y 1) 2 connecting the Hessian with the quartic and its first and second polars; now a, a root of f, is also a root of Ox, and con se uentl the first polar 1 of of q y p f?

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  • It dissolves in an excess of alkali to form plumbites of the general formula Pb(OM) 2.

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  • Then, from the formula a 3 - Mo - [20.12773] - n - 15.59814 Log.

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  • When either the magnetization I or the induction B corresponding to a given magnetizing force H is known, the other may be found by means of the formula B = 41rI + H.

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  • Steinmetz's formula may be tested by taking a series of hysteresis curves between different limits of B,' measuring their areas by a pianimeter, and plotting the logarithms of these divided by 47r as ordinates against logarithms of the corresponding maximum values of B as abscissae.

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  • Trans., 1893, 184, 1017) have in this manner examined how nearly and within what range a formula of the type W =nB E may be taken to represent the facts.

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  • But though a formula of this type has no physical significance, and cannot be accepted as an equation to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a formula may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.

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  • It has been shown by Kennelly (Electrician, 1892, 28, 666) that Steinmetz's formula gives approximately correct results in the case Of nickel.

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  • Shimizu 3 indicate that Steinmetz's formula holds for nickel and annealed cobalt up to B =3000, for cast cobalt and tungsten steel up to B =8000, and for Swedish iron up to B =18,000, the range being in all cases extended at the temperature of liquid air.

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  • It is, of course, true for permanent magnets, where H = o, since then F = 27rI 2; but if the magnetization is due to electric currents, the formula is only applicable in the special case when the mutual action of the two magnets upon one another is supplemented by the electromagnetic attraction between separate magnetizing coils rigidly attached to them.2 The traction method was first employed by S.

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  • He showed that there was, on the whole, a fair agreement between the values determined ballistic ally and those given by the formula B = 871-F.

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  • 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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  • The range of + B within which Steinmetz's formula is applicable becomes notably increased at low temperature.

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  • Steinmetz's formula applies only for very weak inductions when the alloys are at the ordinary temperature, but at the temperature of liquid air it becomes applicable through a wide range of inductions.

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  • Owing to the difficulty of determining the magnetization I and the susceptibility K with accuracy, it has not yet been possible to submit this formula to a quantitative test, but it is said to afford an indication of the results given by actual experiment.

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  • It is pointed out that this formula may be used as a temperature correction in magnetic determinations carried out in air.

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  • His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical formula of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities.

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  • To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.

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  • A formula was found by Bynkershoek in his De dominio maxis (1702) for the restriction of dominion over the sea to the actual distance to which cannon range could protect it.

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  • ALDEHYDES, a class of chemical compounds of the general formula R�CHO (R=an alkyl or an aryl group).

    0
    0
  • ALUM, in chemistry, a term given to the crystallized double sulphates of the typical formula M 2 SO 4 � MP' �(S04) 324H20, where M.

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  • And accordingly it gave rise to the three great doctrines which divided the medieval schools: Realism of the Platonic type, embodied in the formula universalia ante rein; Realism of the Aristotelian type, universalia in re; and Nominalism, including Conceptualism, expressed by the phrase universalia post rem, and also claiming to be based upon the Peripatetic doctrine.

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  • The inherent difficulties of Realism led to a J variety of attempts to reach a more satisfactory formula.

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  • Biot, who assisted in the correction of its proof sheets, remarked that it would have extended, had the demonstrations been fully developed, to eight or ten instead of five volumes; and he saw at times the author himself obliged to devote an hour's labour to recovering the dropped links in the chain of reasoning covered by the recurring formula.

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  • In the case of division we get two kinds of arithmetical formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.

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  • In the second place, particular cases lay the foundation for the general formula.

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  • In this, as in almost all other cases, particular examples should be worked before obtaining a general formula.

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  • a 7; and from this we easily pass to the general formula for the sum of a geometrical progression having a given number of terms.

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  • Thus a comparison of numbers occurring in a table of squares 1 2 =1 112=121 2 2= 4 122=144 3 2 =9 132=169 suggests the formula (A+a)2=A2+2Aa+a2.

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  • The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.

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  • (i.) For the general formula, we need only consider (A+a)".

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  • Now suppose that the formula (2) has been established for every power of A+a up to the (n-i)th inclusive, so that (ii_ I) = (n- I) (r), (y I) = (n -1) (r _ l).

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  • Hence the formula (2) is also true for the nth power of A-I-a.

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  • (vii.) In order to make the formula (5) hold for the extreme values n (o) and n (n) we must adopt the convention that o !

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  • Comparing this with the table in �43 (iv.), and with formula (21), we see that the series expressing the sum may be regarded as consisting of two, viz.

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  • Thus we get a formula for the sum of n terms of a series such as 2.4.6+4.6.8+..., or 6.8.10.12+8.10.12.14+...

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  • The general formula (which is established by more advanced methods) is 4.

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  • The binomial theorem may, for instance, be stated for (x+a)n alone; the formula for (x-a)" being obtained by writing it as {x+(-)a} n or Ix+(- a) } n, so that (x-a) n =x"- 1)xn-laF...+(-)rn(r)xn-rar+..., where + (-) r means - or + according as r is odd or even.

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  • (ii.) One result of these laws is that, when we have obtained any formula involving a letter a, we can replace a by a multinomial.

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  • in � 46 (iii.) is 18(n3)[2]; formula (20) of � 43 (iv.) holds for the extended coefficients, and therefore the sum of n terms of this series is 18.(n-3) [3] -18.

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  • Whatever value is to be attached to Mendel's observation of the breaking up of self-fertilized hybrids of cultivated varieties into the two original parent forms according to the formula " 'PP, 2PN, INN," it cannot be considered as more than a contribution to the extensive investigation of heredity which still remains to be carried out.

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  • 15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula isin 2 u f sin u du = du =7-.

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  • From the general formula (2), if A be the area of aperture, 102 = A2 / x2 f (7) The formation of a sharp image of the radiant point requires that the illumination become insignificant when, n attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture.

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  • The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory.

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  • Fresnel (1815), who gave a formula equivalent to (5) below.

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  • hen d =a the general formula becomes sin' Zm7r Bm: B = (3), showing that, when m is even, B m vanishes, and that, when m is odd, B m: B =1/m272.

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  • 2 m - 1 m 2sin ' a formula which may be verified by Fourier's theorem.

    0
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  • If 8 and 4' denote the angles with the normal made by the incident and diffracted rays, the formula (5) still holds, and, if the deviation be reckoned from the direction of the regularly reflected rays, it is expressed as before by (0+0), and is a minimum when 8 = 0, that is, when the diffracted rays return upon the course of the incident rays.

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  • nor diminishes the If we define as the " dispersion " in a particular part of the spectrum the ratio of the angular interval dB to the corresponding increment of wave-length dX, we may express it by a very simple formula.

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  • This is the ordinary formula for a reflecting plane grating, and it shows that the spectra are formed in the usual directions.

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  • Taking 2 rv2 = u (9), we may write (17r oueiudu ?/ 2) u Again, by a known formula, 1 1 °° -1 uu = 1/ 7r?r o Substituting this in (to), and inverting the order of integration, we get uc 2?

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  • The integration of the several terms may then be effected by the formula e y dy =r(4+2)=(4 - i)(4-2)...

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  • Peligot's results, though called in question by Berzelius, have been amply confirmed by all subsequent investigators; only now, on theoretical grounds, first set forth by Mendeleeff, we double Peligot's atomic weight, so that U now signifies 240 parts of uranium, while UO 3 stands as the formula of the yellow oxide, and UO 2 as that of Berzelius's metal.

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  • The above formula for piperic acid was confirmed by its synthesis by A.

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  • Stannic sulphide, SnS 2, is obtained by heating a mixture of tin (or, better, tin amalgam), sulphur and sal-ammoniac in proper proportions in the beautiful form of aurum musivum (mosaic gold) - a solid consisting of golden yellow, metallic lustrous scales, and used chiefly as a yellow "bronze" for plaster-of-Paris statuettes, &c. The yellow precipitate of stannic sulphide obtained by adding sulphuretted hydrogen to a stannic solution readily dissolves in solutions of the alkaline sulphides to form thiostannates of the formula M 2 SnS 31 the free acid, H2SnS3, may be obtained as an almost black powder by drying the yellow precipitate formed when hydrochloric acid is added to a solution of a thiostannate.

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  • This colouring power is due to the presence of polychlorite, a substance whose chemical formula appears to be C4 8 1-160018, and which may be obtained by treating saffron with ether, and afterwards exhausting with water.

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  • The smallest empirical formula for cellulose (q.v.) may certainly be written C6H1005.

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  • Soc., 1901, 23, p. 527), who, assuming the formula C24H4 6 O 20 for cellulose, showed how the nitrocelluloses described by different chemists may be expressed by the formula C24H4,..-z020(N02)s, where x has the values 4, 5, 6, ...

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  • In the recent methods the cotton remains in contact with the acids for two to four hours at the ordinary air temperature (15° C.), in which time it is almost fully nitrated, the main portion, say 90%, having a composition represented by the formula e C6H702(N03)3, the remainder consisting of lower nitrated products, some oxidation products and traces of unchanged cellulose and cellulose sulphates.

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  • Assuming the above formula to represent guncotton, there is sufficient oxygen for internal combustion without any carbon being left.

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  • If guncotton be correctly represented by the formula C 6 H 7 0 2 (NO 3) 3, it should contain a little more than 14% of nitrogen.

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  • TRIAZOLES (pyrro-a and (3'-diazoles), in organic chemistry, a series of heterocyclic compounds containing the ring complex (annexed formula).

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  • Busch (Ber., 1905, 38, pp. 856, 4049) has isolated a series of bridged ring compounds which he describes as endo-iminodihydrotriazoles, the triphenyl derivative (annexed formula) being prepared by condensing triphenylaminoguanidine with formic acid.

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  • In reply to Rudolf Hospinian's Concordia discors (1607), he wrote a work, rich in historical material but one-sided in its apologetics, Concordia concors (1614), defending the formula of Concord, which he regarded as inspired.

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  • Another work, based on the formula of Concord, was entitled Loci communes theologici.

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  • His formula was "to royalize France and to nationalize the monarchy."

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  • of the metal, taking water as 800 times denser than air on the average, in round numbers, and formula (to) may be written n tan 6=ir, or n6=180, when 6 is a small angle, and given in degrees.

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  • From this formula (26) the table following has been calculated by A.

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  • I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.

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  • Its dental formula (as that of Tragulus) is i., c. m.1 =34.

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  • substances of the general formula Cn(H20)m.

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  • The sugars obtained from honey were investigated by Lowitz and Proust, and the latter decided on three species: (I) cane sugar, (2) grape sugar, and (3) fruit sugar; the first has the formula C12H2201,, the others C 6 H 12 0 6.

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  • The first term includes simple sugars containing two to nine atoms of carbon, which are known severally as bioses, trioses, tetroses, pentoses, hexoses, &c.; whilst those of the second group have the formula C12H22011 and are characterized by yielding two monosaccharose molecules on hydrolysis.

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  • In addition trisaccharoses are known of the formula C13H32016; these on hydrolysis yield one molecule of a monosaccharose and one of a disaccharose, or three of a monosaccharose.

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  • The disaccharoses have the formula C12H22011 and are characterized by yielding under suitable conditions two molecules of a hexose: C12H22011+H20=C6H1206+C6H1206.

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  • Croce's substitutes for the old formula " system " the new formula " systematization."

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  • His formula was modified by Gadamer (Abs.

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  • This formula, which necessitates the presence of two asymmetric carbon atoms in an alkyl tetrahydroberberine, has been accepted by M.

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  • According to Mayer, the formula of convolvulin is C34HSO016, and that of jalapin C31HS0016.

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  • Savonarola's programme of the new government was comprised in the following formula: - (r) fear of God and purification of manners; (2) promotion of the public welfare in preference to private interests; (3) a general amnesty to political offenders; (4) a council on the Venetian model, but with no doge.

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  • To the bishop's formula, "I separate thee from the church militant and the church triumphant," Savonarola replied in firm tones, "Not from the church triumphant; that is beyond thy power."

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  • C. von Hahn as the Aryan Expulsion and Return formula, which counts among its representatives such heroes as Perseus, Cyrus, Romulus and Remus, Siegfried, and, as Alfred Nutt has pointed out, Arthur himself.

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  • Spencer tries to express in a sweeping general formula the belief in progress which pervaded his age, and to erect it into the supreme law of the universe as a whole.

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  • The Formula of Concord (1577),which gave to the whole Lutheran Church of Germany a common doctrinal system, declined to accept the Calvinistic position that man's condemnation as well as his salvation is an object of divine predestination.

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  • Methane and its homologues give origin to the " paraffin " or " fatty series " of the general formula C,H 2, ,+ 1 000H, ethylene gives origin to the acrylic acid series, C n H 27, - 1 000H, and so on.

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  • Dibasic acids of the paraffin series of hydrocarbons have the general formula C n H 2 (000H) 2 "; malonic and succinic acids are important members.

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  • By uttering a sacred formula the good spirit throws the evil one into a state of confusion for a second 3000 years, while he produces the archangels and the material creation, including the sun, moon and stars.

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  • The summit C of the branch BCD occurs at a percentage exactly corresponding to the formula Hg 2 T1.

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  • Sir Edward Frankland,showed how it could be derived from, and converted into, ethane; and thus determined it to be ethane in which one hydrogen atom was replaced by a hydroxyl group. Its constitutional formula is therefore CH3�CH2.OH.

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  • The "quantity at proof" is given by the formula: - quantity of sample X (degrees over or under proof + ioo) divided by ioo.

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  • Potassium and sodium readily dissolve in ethyl alcohol with the production of alcoholates of the formula C2 H5 OK(Na).

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  • In the extreme case when e=1, the prolate ellipsoid becomes a long thin rod, and then the capacity is given by C 1 = a/log e 2a/b (io), which is identical with the formula (2) already obtained.

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  • This formula is important in connexion with the capacity of electric cables, which consist of a cylindrical conductor (a wire) enclosed in a conducting sheath.

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  • Kirchhoff's expression is as follow d+47 r rd l dlog e 167x 2 + t), +t log,: t t I (4) In the above formula e is the base of the Napierian logarithms. The first term on the right-hand side of the equation is the expression for the capacity, neglecting the curved edge distribution of electric force, and the other terms take into account, not only the uniform field between the plates, but also the non-uniform field round the edges and beyond the plates.

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  • 2 If a large plate has a Guard circular hole cut in it, and this is nearly filled up by a circular plate lying in the same plane, and if we place another large plate parallel to the first, then the electric field between this second plate and the small circular plate is nearly uniform; and if S is the area of the small plate and d its distance from the opposed plate, its capacity may be calculated by the simple formula C=S/47rd.

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  • Many complicated expressions have been suggested by subsequent writers in the attempt to represent the continuity of the gaseous and liquid states in a single formula, but these are of a highly empirical nature, and beyond the scope of the present inquiry.

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  • Zeuner's formula for steam), but they cannot be made to represent with sufficient approximation the deviations from the ideal state at moderate pressures and generally lead to erroneous results.

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  • This simple result is generally true, and the corresponding expressions for G" and J" are valid, provided that c - b in formula (17) is a function of the temperature only.

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  • They are not conceived of in any anthropomorphic form, their sex even may often be indeterminate ("sive mas, sive femina" is the constantly recurring formula of prayer), but the sphere of action of each is clearly marked and an appeal to a spirit outside his own special sphere would never even be thought of.

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  • It is necessary in the first place to make quite certain that the right deity is being addressed: hence it is well to invoke all the spirits who might be concerned, and even to add a general formula to cover omissions: here we have the ritual significance of the indigitamenta.

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  • All-important, too, is the order of ceremonial and the formula of prayer: a mistake or omission or an unpropitious interruption may vitiate the whole ritual, and though such misfortunes may occasionally be expiated by the additional offering of a piaculum, in more serious cases the whole ceremony must be recommenced ab initio.

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  • In conjunc tion with Andrea and Selnecker he induced the Lutherans of Saxony and Swabia to adopt the Formula Concordiae and so become one body.

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  • 3H20, but it may be an ammine of the formula [Au(NH 3) 2 (OH) 2 ]OH.

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  • Whether the first creed of the primitive Church was of the simple Christological character which confession of Jesus as the Lord expresses, or of an enlarged type based on the baptismal formula (Matt.

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  • 2) supplies only parallels to the baptismal formula (Matt.

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  • Zahn acutely suggests that it was omitted in the time of Zephyrinus to counteract Monarchian teaching such as the formula: " believe in one God, Jesus Christ."

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  • Schmalkalden, drawn up by Luther in 1536, Luther's catechisms, and the Formula of Concord which was an attempt to settle doctrinal divisions promulgated in 1580, sum up what is called " the confessional theology of Lutheranism."

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  • It shows the influence of Arminian theology against Calvinism, which was vigorously upheld in the Quin-particular formula, put forward by the synod of Dort in 1619 to uphold the five points of Calvinism, after heated discussion, in which English delegates took part, of the problems of divine omniscience and human free-will.

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  • True, the impression conveyed by the attitude of the Catholic party at the second Diet of Spires had served to awaken the feeling for solidarity among the Evangelicals there assembled; and on the 22nd of April they had even secured the basis for a provisional alliance in the shape of a formula drawn up by Bucer and dealing with the Lord's Supper.

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  • With the experience thus gained in manipulating the vacuum, the achievement of thoroughly verifying the pressure of radiation on both opaque and transparent bodies, in accordance with Clerk Maxwell's formula, has been effected (Physical Review, 1901, and later papers) by E.

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  • The dental formula is i.

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  • He found out the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines.

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  • This formula must have been known to Vieta in 1593.

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  • The vapour density is calculated by the following formula: D - W(1 +at) X587,780 (p-s) V in which W =weight of substance taken, V =volume of air expelled, a= 1/273 = .003665, t and p = temperature and pressure at which expelled air is measured, and s= vapour pressure of water at 1°.

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  • There are several corrections of the formula O =W/(W - W1) necessary to the accurate expression of the density.

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  • The pipette having been carefully dried, the process is repeated with pure alcohol or with proof spirits, and the strength of any admixture of water and spirits is determined from the corresponding number of drops, but the formula generally given is not based upon sound data.

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  • The "service of God and his majesty" was the formula which expressed the belief of the sovereign and his subjects.

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  • He has been classed with Arians, but he endorses in his own way the homoousian formula, and denounces Arius as " Christi gloriae incapacissimus."

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  • Such a simple formula is only possible because the salts of sea-water are of such uniform composition throughout the whole ocean that the chlorine bears a constant ratio to the total salinity as newly defined whatever the degree of concentration.

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  • The temperature of maximum density of sea-water of any specific gravity was found by Knudsen to be given with sufficient accuracy for all practical purposes by the formula 0 = 3.950.2660 -0, where 0 is the temperature of maximum density in degrees centigrade.

    0
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  • Hansen the freezing-point T ° C. varies with the degree of concentration according to the formula T = - 0.0086-0 0064-6330"o - o 0001055a02.

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  • The relation of the elevation of the boiling-point (t°) to the osmotic pressure (P) is very simply derived from the formula t=o 02407P 0, while the reduction of vapour pressure proportional to the concentration can be very easily obtained from the elevation of the boiling-point, or it may be obtained directly from tables of vapour tension.

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  • Ruppin's formula as L=o oo1465 S - o o0000978 S 2 +0.0000000876 S 3 in reciprocal ohms.

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    0
  • ACETYLENE, klumene or ethine, a gaseous compound of carbon and hydrogen, represented by the formula C 2 H 2.

    0
    0
  • (5) If c is the resultant velocity of a molecule, so that c 2 =u2+v2+w2, it is readily found from formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is 4lrs1,l (h 3 rn 3 17r 3)e hmc2 c 2 dc. (6) These formulae express the " law of distribution of velocities " in the normal state: the law is often called Maxwell's Law of Distribution.

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  • The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between u and u+du, v and v+dv, w and w+dw, will, by formula (5), be v?l (h 3 m 3 /7 3)e hm (u2+v2+w2)dudvdw (9) per unit volume.

    0
    0
  • If in formula Molecular (?3) we put p=I 013X10 6, (v +v'+ ..

    0
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  • If p is the density corresponding to pressure p, we find that,}, formula (Ii) assumes the form P = 3PC2, where C is a velocity such that the gas would have its actual translational energy if each molecule moved with the same velocity C. By substituting experimentally determined pairs of values of p and p we can calculate C for different gases, and so obtain a knowledge of the magnitudes of the molecular velocities.

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  • - The pressure as given by formula (12) can be written as the sum of a number of separate terms, one for each gas in the mixture.

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  • - From formula (13) it appears that v+v'+..

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  • (21) The comparison of this formula with experiment provides a striking confirmation of the truth of the kinetic theory but at the same time discloses the most formidable difficulty which the theory has so far had to encounter.

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  • On giving different values to n in formula (21), we obtain the values for y: n = o, 1, 2, 3, 4, 5, y = 1.66, 1.5, 1.4, 1.33, 1.28, 1.25, &c.

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  • To examine what is meant by a zero value of n we refer to formula (15).

    0
    0
  • If a solid body is regarded as an aggregation of similar atoms each of mass m, its specific heat C is given, as in formula (19) by C = i (n+3) R/Jm.

    0
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  • In the formula used by Synesius (4,0) which is to be found in Bingham's Antiquities, we already find the attention of magistrates specially called to the censured person.

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  • With it may be compared the formula transcribed by Dr Burton in his History of Scotland (iii.

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  • in penance, of the formula " I absolve thee "; in the Eucharist, of the words " This is my body " and " This is the cup of my blood " or " This is my blood "; in confirmation, of the words " I sign thee with sign of the cross and confirm thee with chrism of salvation in name of Father and Son and Holy Spirit "; in baptism, of the words " I baptize thee in the name of Father, Son and Holy Spirit (or among the Greeks " N.

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  • In conditional baptism the Latins, since about the year 1227, use the formula, " If thou art not baptized, then do I baptize thee," &c. The Latins further insist on a strict observance of the traditional matter and form.

    0
    0
  • Its compound with calcium chloride has the formula CaC1 2.4CH 3.

    0
    0
  • It passes over equal spaces in equal times, but falls with an accelerating velocity according to the formula h = zgt 2, where h is the height fallen through, g the force of gravity, and t the time of flight.

    0
    0
  • The formula for length of scale is, length = sighting radius X tangent of the angle of elevation.

    0
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  • HH' = 60 X 60' the resultant angle of deflection is HFH', and this can be determined by the same formula a = h X1200 X 3, but in this case h = HH' = R R 60 60 R X 3600 a _ = i' so that if the sight is inclined to the left I° it will R X 3600 give 1' deflection for every degree of elevation.

    0
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  • BCDcanalwaysbe cal culated from the formula, angle of sight in minutes - (in feet) X 1146 R (in yards) (h = height of gun above sea-level; R =range).

    0
    0
  • None are to partake of it save those who have been "baptized in the name of the Lord" (an expression which is of interest in a document which prescribes the threefold formula).

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  • 410) On the Apostles' Creed, composed that formula by contributing each a single sentence.

    0
    0
  • On the Protestant side the identity is still clear in the Lutheran Formula of Concord (1577).

    0
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  • "Dogmengeschichte " in HerzogHauck's Realencykl., 1898) that dogma is historically equivalent to regula fidei, he is in flat contradiction to the " dogma " of his own church as stated in the Formula of Concord.

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  • It has the exact vapour-density corresponding to the formula.

    0
    0
  • But an area or a volume cannot generally be measured by successive applications of the unit of area or volume; intermediate processes are necessary, the result of which is expressed by a formula.

    0
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  • the direct measurement of certain magnitudes (usually lengths) in terms of a unit, and the application of a formula for determining the area or volume from these data.

    0
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  • Mensuration, then, is mainly concerned with quadratureformulae and cubature formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume.

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  • The deduction of the formula tab, where a and b are numbers, should be regarded as a later step.

    0
    0
  • This involves the use of Cartesian co-ordinates, and leads to important general formulae, such as Simpson's formula.

    0
    0
  • The numerical result obtained by applying a formula to particular data will generally not be exact.

    0
    0
  • (i) The formula itself may not be numerically exact.

    0
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  • (a) The formula may involve numbers or ratios which cannot be expressed exactly in the ordinary notation.

    0
    0
  • (b) The formula may only be approximative.

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  • The length of the arc of a circle, for instance, is known if the length of the chord and its distance from the middle point of the arc are known; but it may be more convenient in such a case to use a formula such as Huygens's rule than to obtain a more accurate result by means of trigonometrical tables.

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  • (a) The nature of the bounding curve or surface may not be exactly known, so that certain assumptions have to be made, a formula being then used which is adapted to these assumptions.

    0
    0
  • Such a formula is approximative, in that it is known that the result of its application will only be approximately correct; it differs from an approximative formula of the kind mentioned in (i) (b) above, in that it is adopted of necessity, not by choice.

    0
    0
  • (iii) In dealing with cases where actual measurements are involved, the error (i) due to inaccuracy of the formula will often be negligible in comparison with the error (ii) due to inaccuracy of the data.

    0
    0
  • For this reason, formulae which will only give approximate results are usually classed together as rules, whether the inaccuracy lies (as in the case of Huygens's rule) in the formula itself, or (as in the case of Simpson's rule) in its application to the data.

    0
    0
  • The formula then becomes area =z(MP.NB+NQ.

    0
    0
  • This formula holds for the general case in which the base is a trapezium; the wedge being thus formed by cutting a triangular prism by any two planes.

    0
    0
  • This is known as the prismoidal formula.

    0
    0
  • The formula is a deduction from a general formula, considered later (§ 58), and may be verified in various ways.

    0
    0
  • Hence, by (i) and (ii), the formula holds for this figure.

    0
    0
  • By (i) and (iii), the formula holds for each of these figures; and therefore it holds for the prismoid as a whole.

    0
    0
  • Another method of verifying the formula is to take a point Q in the mid-section, and divide up the prismoid into two pyramids with vertex Q and bases ABCD ...

    0
    0
  • The first of these is a particular case of the prismoidal formula (§ 58).

    0
    0
  • The area is then given by the formula of § 25 (ii).

    0
    0
  • This is Simpson's formula.

    0
    0
  • This is Simpson's second formula.

    0
    0
  • It may be deduced from the formula given above.

    0
    0
  • Simpson's (first) formula, for instance, holds for f = I, and is obtained by taking p = i and ignoring differences after 52up.

    0
    0
  • 'Simpson's second formula is obtained by taking m = 3 and ignoring differences after /,u52u1m.

    0
    0
  • Thus we find from (i) that Simpson's second formula, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us).

    0
    0
  • This might have been deduced directly from Simpson's first formula, by a series of eliminations.

    0
    0
  • The same formula will also hold (§ 52) for any cubic trapezette through the points.

    0
    0
  • It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's formula may be applied to find the volume of the solid.

    0
    0
  • This formula applies to such figures as the cone, the sphere, the ellipsoid and the prismoid.

    0
    0
  • The formula is known as the prismoidal formula.

    0
    0
  • 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+..

    0
    0
  • This ordinate will be an algebraical function of x, and we can again apply a suitable formula.

    0
    0
  • The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second formula; call these Ao and A2, and Al.

    0
    0
  • Hence Simpson's formula applies, and the volume is lh(A0-I-4A1+ A2).

    0
    0
  • The process is simplified by writing down the general formula first and then substituting the values of u.

    0
    0
  • The formula, in the above case, is 3h{ *k(uo,o + 3 where u 0, 0 denotes the ordinate for which x=xo+Oh, y=yo+c¢k The result is the same as if we multiplied lk(vo 3v1+3v2 + v 3) by lh(uo 4u1 +u2), and then replaced uovo, uov1, ..

    0
    0
  • The volume of a frustum of a cone, for instance, can be expressed in terms of certain magnitudes by a certain formula; but not only will there be some error in the measurement of these magnitudes, but there is not any material figure which is an exact cone.

    0
    0
  • The formula may, however, be used if the deviation from conical form is relatively less than the errors of measurement.

    0
    0
  • In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular formula differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.

    0
    0
  • Thus a quadrature-formula is a formula for expressing [A x .24] or fudx in terms of a series of given values of u, while a cubature-formula is a formula for expressing [[Vx, 0 .

    0
    0
  • Applying Simpson's formula to each of these, we have A -__9= *h(uo + 4 14 1 + u2) + 3 h (14 2 + 4u3 + u4) + ..

    0
    0
  • (ii) Similarly, if m is a multiple of 3, the repeated application of Simpson's second formula gives Simpson's second rule A 1?

    0
    0
  • The legitimacy of this assumption, and of the further assumption which enables the area of the new figure to be expressed by an approximate formula instead of by an exact formula, must be verified in every case by reference to the actual differences.

    0
    0
  • (iv) In order to find what formula may be applied, it is necessary to take the successive differences of u; and it is then just as easy, in most cases, to use a formula which directly involves these differences and therefore shows the degree of accuracy of the approximation.

    0
    0
  • The alternative method, therefore, consists in taking a simple formula, such as the trapezoidal rule, and correcting it to suit the mutual relations of the differences.

    0
    0
  • Their generalization is given by the Euler-Maclaurin formula = I, = 0, = 0, = 0 .

    0
    0
  • x=xo, and an analogous formula (which may be obtained by substituting 2h and C, for h and C 1 in the above and then expressing T 1 as 2C -CO) A =J xo udx=T1+ [4h2u'_+ 15487$5 h8uvii ?

    0
    0
  • of x, the formula of (i) applies, provided we take (x)=(I - 24 62 +04?1 s4 -1-h-06 +

    0
    0
  • The generalized formula is fxo u¢ (x)dx = Aq(xm) - where 0, A 2, ...

    0
    0
  • are found from the corresponding coefficients in the; uler-Maclaurin formula (§ 75) by multiplying them by Pp2k+(1g2k+Rr2k..

    0
    0
  • 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.

    0
    0
  • The Euler-Maclaurin formula (§ 75) assumes that the bounding values of u', u"',..

    0
    0
  • To find the quantity of timber in a trunk with parallel ends, the areas of a few sections must be calculated as accurately as possible, and a formula applied.

    0
    0
  • The formula applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56).

    0
    0
  • - FOr applications of the prismoidal formula, see Alfred Lodge, Mensuration for Senior Students (1895).

    0
    0
  • He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p. His value of the velocity in air is therefore U = iJ (p ip.) (Newton's formula).

    0
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  • That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence d p /d p = y k p Y 1= yp/p, and U = (yp/p) [Laplace's formula].

    0
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  • This formula was very nearly confirmed for hydrogen, carbon dioxide and nitrous oxide.

    0
    0
  • tube by Kirchhoff's formula, Violle and Vautier found for the velocity in open air at o° C.

    0
    0
  • The velocity deduced at 8.1° C. was U=1435 met./sec., agreeing very closely with the value calculated from the formula U 2 = E/p.

    0
    0
  • Kundt also obtained results in general agreement with the formula (Rayleigh, Sound, ii.

    0
    0
  • AMEN, a Hebrew word, of which the root meaning is "stability," generally adopted in Christian worship as a concluding formula for prayers and hymns.

    0
    0
  • Its introduction into the baptismal formula (in the Greek Church it is pronounced after the name of each person of the Trinity) is probably later.

    0
    0
  • Then w2'/w2 = (1 +4p7p) w2'/7.vl = 1 1 0 [l2/ll + (12/11) 2] (1 +4P1p) A partially rational approximate formula for the weight of main girders is the following (Unwin, Wrought Iron Bridges and Roofs, 1869, p. 4 0) :- Let w=total live load per ft.

    0
    0
  • Hence the formula is more useful in the form w = (w i +w2)1 2 / (Kd -1 2) = (wl +w 2)lr/ (K -lr) where k= (wl+w2-1-w3)lr/w3 is to be deduced from the data of some bridge previously designed with the same working stresses.

    0
    0
  • The following table gives values of the working stress calculated by these equations: Working Stress for Tension or Thrust by Launhardt and Weyrauch Formula.

    0
    0
  • run in tons, is wa = (wl+w2)1r/(K-lr) according to a formula already given, then w 3 becomes infinite if k-lr =o, or if 1 = K/r, F I --xJ- ----- ---- -?

    0
    0
  • The reciprocal figure for any loaded frame is a complete formula for the stress on every member of a frame of that particular class with loads on given joints.

    0
    0
  • About 15% of a volatile oil is obtained by distilling cubebs with water; after rectification with water, or on keeping, this deposits rhombic crystals of camphor of cubebs, C 15 H 26 O; cubebene, the liquid portion, has the formula C15HV4.

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    0
  • It is evident, therefore, that the request for a definition of Ultramontanism cannot be answered with a concise formula, but that the varied character of its manifestations necessitates a more detailed examination of its peculiar objects.

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    0
  • 16, &c. In primitive times, when confession of sins was made before the congregation, the absolution was deferred till the penance was completed; and there is no record of the use of any special formula.

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  • But about the r3th century the Roman formula was altered, and the council of Trent (1551) declared that the "form" and power of the sacrament of penance lay in the words Ego to absolvo, &c., and that the accompanying prayers are not essential to it.

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  • and so the divine righteousness was satisfied - a formula, however, which left much room for contro versy.

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    0
  • On hearing from Rome, Cyril at once held a synod and drew up a doctrinal formula for Nestorius to sign, and also twelve anathemas covering the various points of the Nestorian dogmatic. Nestorius, instead of yielding to the combined pressure of his two great rivals, merely replied by a counter excommunication.

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  • The commotion which had been thus raised did not so easily subside in the more eastern section of the church; the Antiochenes continued to maintain for a considerable time an attitude of antagonism towards Cyril and his creed, and were not pacified until an understanding was reached in 433 on the basis of a new formula involving some material conce9sions by him.

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  • Silberrad assigns the formula NH 3 �NI 3 to the compound, and explains the decomposition as taking place, 2NH3�N13+ 6Zn (C2H5)2 = 6ZnC 2 H 5 �I+2NH 3 +2N(C 2 H 5) 3.

    0
    0
  • Soc., 1889, 55, p. 163) determined the vapour density of hydrofluoric acid at different temperatures, and showed that there is no approach to a definite value below about 88° C. where it reaches the value 10.29 corresponding to the molecular formula HF; at temperatures below 88° C. the value increases rapidly, showing that the molecule is more complex in its structure.

    0
    0
  • His seeming inconsistencies are reconciled to apprehension, not by a formula of the intellect, but by the many-sidedness of a highly impressible nature.

    0
    0
  • MacCullagh's hands the correct equations were derived from a single energy formula by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming.

    0
    0
  • The polarization itself is determined from the electric force (P,Q,R) by the usual statical formula of linear type which becomes tor an isotropic medium (.f',g',h') = c2(P,Q,R), because any change of the dielectric constant K arising from the convection of the material through the aether must be independent of the sign of v and therefore be of the second order.

    0
    0
  • Legendre's formula x: (log x- I.

    0
    0
  • Hexahydropyrazine or piperazine (formula IV.

    0
    0
  • In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v m, the integration can be effected which replaces the summation in (to), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f (v)or v m lk or its equivalent Cr, where r is the retardation.

    0
    0
  • Integrating (27) again, (31) y =g(zTt2t 2) = zgt(T -t); and denoting T-t by t', and taking g= 32f/s2,) y =16tt', (32 which is Colonel Sladen's formula, employed in plotting ordinates.

    0
    0
  • To share the S between and 0, the vertex A is taken as the point of half-time (and therefore beyond half-range, because of the continual diminution of the velocity), and the velocity vo at A is calculated from the formula (38) T(vo) = T(V) - = z {T(V) -T (v)}; and now the degree table for D(v) gives (39) 4=C{D(V)-D(v0)}, =CID (vo) -D(v)}.

    0
    0
  • The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article Armour Plates.

    0
    0
  • But no formula has yet been invented, derived on theoretical principles from the physical data, which will assign by calculation a definite magnitude to 3.

    0
    0
  • The formula shows that except for numbers of the form (3n 2 n) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity.

    0
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  • 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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  • The simple olefines containing one doublylinked pair of carbon atoms have the general formula (CnH2n; the di-olefines, containing two doubly-linked pairs, have the general formula C0H2n_2 and are consequently isomeric with the simple acetylenes.

    0
    0
  • The olefines may be synthetically prepared by eliminating water from the alcohols of the general formula CnH2n+1 OH, using sulphuric acid or zinc chloride generally as the dehydrating agent, although phosphorus pentoxide, syrupy phosphoric acid and anhydrous oxalic acid may frequently be substituted.

    0
    0
  • This fact, coupled with the determination of the vapour density of the gas, establishes the molecular formula CO.

    0
    0
  • The second formula, due to R.

    0
    0
  • Potassium at temperatures from 200 to 400°C. occludes hydrogen gas, the highest degree of saturation corresponding approximately to the formula K 2 H.

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    0
  • 25) questions were raised by the occurrence of the formula " until this day " in Jos.

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  • The current theological formula for this two-sided position is that the prophets are at once preachers of the law and forerunners of the gospel; and, as it is generally assumed that they found the law already written, their originality and real importance is made to lie wholly in their evangelical function.

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  • It will be evident even from this rapid sketch, necessarily confined to a few of the most cardinal points, that Hebrew prophecy is not a thing that can be defined and reduced to a formula, but was a living institution which can only be understood by studying its growth and observing its connexion with the historical movements with which its various manifestations were bound up. Throughout the great age of prophecy the most obvious formal character that distinguished it was that the 1 One might say from the days of Habakkuk.

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  • This equation defines log x for positive values of x; if o the formula ceases to have any meaning.

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  • 2 The method consists in the use of the formula sin a sin b=2 {cos(a-b)-cos(a+b)l, by means of which the multiplication of two sines is reduced to the addition or subtraction of two tabular results taken from a table of sines; and, as such products occur in the solution of spherical triangles, the method affords the solution of spherical triangles in certain cases by addition and subtraction only.

    0
    0
  • It will be observed that in the first process the value of the modulus is in fact calculated from the formula.

    0
    0
  • h I Io^ - i - log e 10' the value of la being 1/2 64, and in the second process log i o 2 is in effect calculated from the formula.

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    0
  • (-) 5 + &c., in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another.

    0
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  • From the formula for log e (p/q) we may deduce the following very convergent series for log e 2, log e 3 and log e 5, viz.: log e 2=2(7P +5Q +3R), log e 3 =2(11P+8Q +5R), log e 5=2(16P+12Q+7R), where P 1 +3?1) 3 5 s (31) &C.

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  • The logarithm is then obtained by use of the formula d l d2 l d3 2 log e (x+d) = log e x-f- - x2+3 x3 - &c., in which of course the object is to render dlx as small as possible.

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    0
  • The formula of the famous Jesuit vow is as follows: "I, N., promise to Almighty God, before His Virgin Mother and the whole heavenly host, and to all standing by; and to thee, Reverend Father General of the Society of Jesus, holding the place of God, and to thy successors (or to thee, Reverend Father M.

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  • The vows of this grade are the same as the last formula, with the addition of the following important clause: "Moreover I promise the special obedience to the Sovereign Pontiff concerning missions, as is contained in the same Apostolic Letter and Constitutions."

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  • In the former bull, the pope gives the text of the formula submitted by Ignatius as the scheme of the proposed society, and in it we get the founder's own ideas: ".

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  • We constantly meet with the idea that the soul, on leaving the body, finds its path to the highest heaven opposed by the deities and demons of the lower realms of heaven, and only when it is in possession of the names of these demons, and can repeat the proper holy formula, or is prepared with the right symbol, or has been anointed with the holy oil, finds its way unhindered to the heavenly home.

    0
    0
  • The formula PbS corresponds with lead 86.6 and sulphur 13.4%.

    0
    0
  • The formula arrived at by Cauchy was n= A-FB+?

    0
    0
  • If suitable values are chosen for these constants, the formula can be made to represent the dispersion of ordinary transparent media within the visible spectrum very well, but when extended to the infra-red region it often departs considerably from the truth, and it fails altogether in cases of anomalous dispersion.

    0
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  • There are also grave theoretical objections to Cauchy's formula.

    0
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  • Ketteler and known as the Ketteler-Helmholtz formula, has been much used in calculating dispersion, and expresses the facts with remarkable accuracy.

    0
    0
  • P. Drude has obtained a similar formula based on the electromagnetic theory, thus placing the theory of dispersion on a much more satisfactory basis.

    0
    0
  • As A m is a wave-length corresponding to an absorption band, this formula can be used to find values of A m which satisfy the observed values of n within the region of transparency, and so to determine where the absorption bands are situated.

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    0
  • At George's instigation the second Sirmian formula (promulgated by the third council of' Sirmium 3S7), which was conciliatory towards strict Arianism, was opposed at the council of Ancyra in 358 (Harnack, Hist.

    0
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  • His refusal to subscribe unconditionally to the rigid formula of belief adopted by the theologians of Tubingen permanently closed against him the gates of his alma mater.

    0
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  • Rohner (Ber., 1909, 42, p. 4 0 93) obtained a yellowish white oxide, of the formula I 4 0 9, which they regard as an iodate of tervalent iodine, Millon's oxide being considered a basic iodate.

    0
    0
  • On the next topic of importance, the primacy of the pope, the project of union nearly suffered shipwreck; but here a vague formula was finally constructed which, while acknowledging the pope's right to govern the church, attempted to safeguard as well the rights of the patriarchs.

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  • In another department of physical chemistry he investigated the expansion of liquids with heat, and devised a formula for its expression similar to Gay-Lussac's law of the uniformity of the expansion of gases, while so far back as 1861 he anticipated T.

    0
    0
  • COOH, the first member of the series of aliphatic monobasic acids of the general formula C.H 2, ,O 2.

    0
    0
  • The Fihrist reckons seven principal works of Mani, six being in the Syriac and one in the Persian language; regarding some of these we also have information in Epiphanius, Augustine, Titus of Bostra, and Photius, as well as in the formula of abjuration (Cotelerius, PP. Apost.

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  • And although the God of light himself now took to the field, and with the help of new aeons (the spirit of life, &c.) inflicted total defeat upon Satan, and set the ' A is spoken of in the formula of abjuration, and an Epistola ad virginem Menoch by Augustine.

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  • the sent forth) of Jesus the friend in the love of the Father, of God."He uses the formula: Praise and laud to the Father and the Son and the Holy Spirit."

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  • The milk-dentition, and even the early condition of the permanent dentition, is formed on the same general type as that of Sus, except that certain teeth are absent, the formula being 13 i cl, total 34; but as age advances all the teeth have a tendency to disappear, except the canines and the posterior molars, but these, which in some cases are the only teeth left in the jaws, attain an extraordinary development.

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  • Again, like the hierophant, the sycophant publicly pronounced the formula of exclusion of certain unworthy persons from the celebration of the mysteries of the fig.

    0
    0
  • A synopsis of the leading formula connected with the circle will now be given.

    0
    0
  • Approximate formulae are: - Arc=3(8c 2 - c) (Huygen's formula); arc = s (c - 40c, +256c4).

    0
    0
  • Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a " formula."

    0
    0
  • Accordingly, we find in Vieta a formula for the ratio of diameter to circumference, viz.

    0
    0
  • 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.

    0
    0
  • 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.

    0
    0
  • 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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  • They Are Therefore Connected With The Golden Numbers By The Formula (I), In Which N Is Any Whole Number; Andor A Whole Lunar Cycle (Supposing The First Epact To Be 11), They Are As Follows: 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, I, 12, 23, 4, 15, 26, 7, 18, 29.

    0
    0
  • In Order, Therefore, To Render The Subtraction Possible, L Must Be Increased By Some Multiple Of 7, As 7M, And The Formula Then Becomes 7M L X.

    0
    0
  • In This Case, As The Formula Supposes The Intercalation Already Made, The Resulting Letter Is That Which Applies After The 29Th Of February.

    0
    0
  • Thus For 1840 The Formula Gives D; During The First Two Months, Therefore, The Dominical Letter Is E.

    0
    0
  • In Order To Investigate A Formula For The Epact, Let Us Make E=The True Epact Of The Given Year; J =The Julian Epact, That Is To Say, The Number The Epact Would Have Been If The Julian Year Had Been Still In Use And The Lunar Cycle Had Been Exact;, S =The Correction Depending On The Solar Year; M =The Correction Depending On The Lunar Cycle; Then The Equation Of The Epact Will Be E=J S M; So That E Will Be Known When The Numbers J, S, And M Are Determined.

    0
    0
  • Now This Is A Condition Which Will Evidently Be Expressed In General By The Formula N (N 1).

    0
    0
  • Let (C7) W =A, Then For All Years After 1800 The Value Of M Will Beiven By The Formula (C 18 A), G' Y W Therefore, 3 Counting From The Beginning Of The Calendar In 1582, C 15 A ' M = 3 W By The Substitution Of These Values Of J, S And M, The Equation Of The Epact Becomes E C 16 (C 16 (C 15 A) 3 O R () 4 W 3 W.

    0
    0
  • The Above Formula For The Epact Is Given By Delambre (Hist.

    0
    0
  • The Value Of L Is Always Given By The Formula For The Dominical Letter, And P And 1 Are Easily Deduced From The Epact, As Will Appear From The Following Considerations.

    0
    0
  • When, Therefore, E Is Greater Than 23, We Must Add 30 In Order That P And 1 May Have Positive Values In The Formula P =24 E And 1=27 E.

    0
    0
  • P= 54 E 1=57 E, Or (57 E) 7 R' By Substituting One Or Other Of These Values Of P And 1, According A; The Case May Be, In The Formula P = P (L L), We Shall Have P, Orthe Number Of Days From The 21St Of March To Easter Sunday.

    0
    0
  • By Substituting This Number In The Formula For The Dominical Letter, We Have X =1840, C 16 =2, /C 416) W =0, Therefore L =7M 6 1840 460 2 =7M 2292 = 7X328 2292 =2296 2292 = 4 L =4 =Letter D.

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  • From This Formula The Following Table Has Been Constructed: Table Viii.

    0
    0
  • This Formula Gives The Following Rule For Calculating The Date Of The Commencement Of Any Year Of The Hegira, According To The Gregorian Or New Style.

    0
    0
  • Appended to the London edition of the solar and lunar tables are two short tracts - the one on determining longitude by lunar distances, together with a description of the repeating circle (invented by Mayer in 1752), the other on a formula for atmospheric refraction, which applies a remarkably accurate correction for temperature.

    0
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  • iv.) has shown, to the group of tales classified as the Aryan Expulsion and Return formula, found in all Aryan lands.

    0
    0
  • He describes himself when he says, "The student of Christian doctrine, because he strives after exactness of phrase, because he is conscious of the inadequacy of any one human formula to exhaust the truth, will be filled with sympathy fo