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The Formulae.

In the second of these notices he gives the general formulae without demonstrations.

Nef (1894), who proposed the formulae: C: N OH N: CH CH: N O C: N OH, O ?

The following formulae are important: N - CH C =CH/ - CH CH2 CH CH C/ NH CH / N CH2 N / N / N% CH CH H3C CH2 N-fl-Pyridylpyrrol, a0-pyridylpyrrol, nicotine.

the Brahman or superintending priest; the Hotri or reciter of hymns and verses; the Udgatri or chanter; and the Adhvaryu or offerer, who looks after the details of the ceremonial, including the preparation of the offering-ground, the construction of fireplaces and altars, the making of oblations and muttering of the prescribed formulae.

Its author, with a considerable mathematical and mechanical bias, reckoned entirely with the quantity, not with the quality of his units, and relied almost implicitly upon his formulae.

The resistance caused by the wind is very variable, and in extreme cases may double the resistance found from the formulae.

Chemical Formulae.

Before explaining these formulae it will be necessary, however, to consider the differences in combining power exhibited by the various elements.

Rational formulae of a much simpler description than these graphic formulae are generally employed.

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.

In 1882 Claus suggested a combination of his own and Dewar's benzene formulae.

On the centric hypothesis two formulae are possible: (1) due to H.E.

benzenoid rings as represented by the symbols: - Anthracene Phenanthrene In both cases the medial ring is most readily attacked; and various formulae have been devised which are claimed by their authors to represent this and other facts.

The formulae of these substances are: By substituting one or more CH groups in these compounds by nitrogen atoms, ring-systems, collectively known as azoles, result.

Thiophene yields a similar series: isothiazole (only known as the condensed ring, isobenzothiazole), thiazole, diazosulphides, piazthioles, azosulphimes and thiobiazole (the formulae are easily derived from the preceding series by replacing oxygen by sulphur).

Typical formulae are (R denoting 0, S or NH): Isomers are possible, for the condensation may be effected on the two carbon atoms symmetrically placed to the hetero-atom; these isomers, however, are more of the nature of internal anhydrides.

In the case of the amines he decides in favour of the formulae H2C: NH3 2C NH2 H 3 C. NH CH3 H primary, secondary, tertiary.

These formulae, however, only apply to aliphatic amines; the results obtained in the aromatic series are in accordance with the usual formulae.

In the formulae which follow we suppose 1 and l' to represent the latitudes, a and b the co-latitudes (90° - 1 or 90° - l'), and t the difference in longitude between them or the meridian distance, whilst D is the distance required.

The loculi were intact and the epitaphs still in their places, so that " they form a kind of museum, in which the development, the formulae, and the symbolic figures of Christian epigraphy, from its origin to the end of the 3rd or 4th century, can be notified and contemplated, not in artificial specimens as in the Lateran, but in the genuine and living reality of their original condition."

Protestants have condemned these formulae as so much magic, and in this modern science tends to agree with them; but to orthodox Protestants at least Catholics have a perfect right to reply that, in taking this line, they are but repeating the accusation brought by the Pharisees against Christ, viz.

The appendix de Benedictionibus to the Rituale Romanum contains formulae, often of much simple beauty, for blessing all manner of persons and things, from the congregation as a whole and sick men and women, to railways, ships, blast-furnaces, lime-kilns, articles of food, medicine and medical bandages and all manner of domestic animals.

His earliest publications, beginning with A Syllabus of Plane Algebraical Geometry (1860) and The Formulae of Plane Trigonometry (1861), were exclusively mathematical; but late in the year 1865 he published, under the pseudonym of "Lewis Carroll," Alice's Adventures in Wonderland, a work that was the outcome of his keen sympathy with the imagination of children and their sense of fun.

The value of k, however, does not keep constant so satisfactorily in the case of highly dissociated substances, and empirical formulae have been constructed to represent the effect of dilution on them.

The logarithmic formulae for these concentration cells indicate that theoretically their electromotive force can be increased to any extent by diminishing without limit the concentration of the more dilute solution, log c i /c 2 then becoming very great.

Lowry and Armstrong represent these compounds by the following spatial formulae which postulate a y-oxidic structure, and 5 asymmetric carbon atoms, i.e.

one more than in the Fischer formulae.

These formulae are supported by many considerations, especially by the selective CH 2 OH CH20H CH OH CH OH C C H O

The composition of the several species of mica is given by the following formulae, some of which are only approximate.

Recalling the formulae above which connect s P4 and a m, we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively.

From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.

= (A11+A22)n by the substitutions 51 = A l, E1+ï¿½1 2, 52 = A2E1+ï¿½2E2, the umbrae Al, A2 are expressed in terms of the umbrae al, a 2 by the formulae A l = Alai +A2a2, A2 = ï¿½la1 +ï¿½2a2ï¿½ We gather that A1, A2 are transformed to a l, a 2 in such wise that the determinant of transformation reads by rows as the original determinant reads by columns, and that the modulus of the transformation is, as before, (A / .c).

are said to be contragredient when the linear substitutions for the first set are x =A1X+u1Y-}-v1Z-?--..., y = A2X+,u2Y +v2Z ï¿½..., Z = A 3 X +ï¿½3Y -1v 3 Z - -..., and these are associated with the following formulae appertaining to the second set, .`"?.

In conduct they shrank from formulae.

But their constitution is not yet solved, there even being some divergence of opinion as to their empirical formulae.

The second includes definitions of technical terms in common use, together with so much of the elementary theory as is necessary for understanding the experimental work described in subsequent portions of the article; a number of formulae and results are given for purposes of reference, but the mathematical reasoning by which they are obtained is not generally detailed, authorities being cited whenever the demonstrations are not likely to be found in ordinary textbooks.

Although the above useful formulae, (io) to (15), are true only for an infinitely small magnet, they may be practically applied whenever the distance r is considerable compared with the length of the magnet.

From the former we deduce Ho, and from the latter the corresponding value of I, using the formulae Ho = 47rin/l and I - X s, (d (-- 11)2n7rr 2 i where s is the deflection in scale-divisions, n the distance in scaledivisions between the scale and the mirror, and r the radius of the wire.

Its scope may be briefly described as the reduction of the theory of mechanics to certain general formulae, from the simple development of which should be derived the equations necessary for the solution of each separate problem.

He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but Lagrange, by the happy artifice of transferring the origin of coordinates from the centre of the sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally.

It is the ritual of a magician, imbedded in which, and alternating with magic formulae and other occult matter, are a number of invocations and prayers which Dieterich reconstructs as a liturgy in use by the clergy of Mithras between A.D.

The following structural formulae are assigned to these compounds: R CAN OH R.CCN(OK) > O R C N02K NO 2 N(:O) NO nitrolic acid; erythro-salt; leuco-salt.

4to, Paris, 1799) contains methods for calculating the movements of translation and rotation of the heavenly bodies, for determining their figures, and resolving tidal problems; the second, especially dedicated to the improvement of tables, exhibits in the third and fourth volumes (1802 and 1805) the application of these formulae; while a fifth volume, published in three instalments, 1823-1825, comprises the results of Laplace's latest researches, together with a valuable history of progress in each separate branch of his subject.

Theorems and formulae are appropriated wholesale without acknowledgment, and a production which may be described as the organized result of a century of patient toil presents itself to the world as the offspring of a single brain.

To this lofty quality of intellect he added a rare sagacity in perceiving analogies, and in detecting the new truths that lay concealed in his formulae, and a tenacity of mental grip, by which problems, once seized, were held fast, year after year, until they yielded up their solutions.

p. 258), besides constructing satisfactory formulae for the barometrical determination of heights (Mec. cel.

(ii.) In the above case the two different kinds of statement lead to arithmetical formulae of the same kind.

What is more important is to verify particular examples of general formulae.

These formulae are of two kinds: - (a) the general properties, such as m(a+b) = ma+mb, on which algebra is based, and (b) particular formulae such as (x - a) (x+a) = x 2 - a 2 .

(ii.) These algebraical formulae involve not only the distributive law and the law of signs, but also the commutative law.

This increases the extent of the numbers with which we have to deal; but it enables us to reduce the number of formulae.

They are (a+b)-?-c=a+(b+c) (A) (aXb)Xc=aX(bXc) (A') a+b=b+a (c) aXb=bXa (c') a(b c) =ab-Fac (D) (a - b)+b=a (I) (a=b)Xb=a (I') These formulae express the associative and commutative laws of the operations + and X, the distributive law of X, and the definitions of the inverse symbols - and =, which are assumed to be unambiguous.

The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.

All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.

By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...

As in quaternions, so in the extensive calculus, there are numerous formulae of transformation which enable us to deal with extensive quantities without expressing them in terms of the primary units.

These may be compared and contrasted with such quaternion formulae as S(VabVcd) =SadSbc-SacSbd dSabc = aSbcd - bScda+cSadb where a, b, c, d denote arbitrary vectors.

The sum and product of two quantities are defined in the first instance by the formulae zae -IE(3e = E (a +0) e, Za,ei X E ai e j = (a iai) eie9, so that the laws A, C, D of ï¿½ 3 are satisfied.

formulae derived by Stokes (Camb.

(15), formulae which may be utilized for the calculation of G, H when u (or v) is small.

If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered.

C02-}-NH3 the various units of the ring system being illustrated in the annexed formulae: C C C NI^C CI C N/ C A N ` N' N C,C N N N a-Triazines, (-Triazines, Cyanidines.

The oldest and perhaps most reasonable idea represents guncotton as cellulose trinitrate, but this has been much disputed, and various formulae, some based on cellulose as C, 2 H200 10, others on a still more complex molecule, have been proposed.

In the transitional period, when the Arabian school began to influence European medicine, but before the Salernitans were superseded, comes Nicolaus Praepositus, who wrote the Antidotarium, a collection of formulae for compound medicines, which became the standard work on the subject, and the foundation of many later compilations.

The formulae of these substances are: - H2Ni /N 'NJ NH2 Chrysaniline.

It is convenient to treat these glasses as " normal " glasses, but they are in reality mixtures of silicates, and cannot rightly be regarded as definite chemical compounds or represented by definite chemical formulae.

Sodium percarbonates of the formulae Na 2 CO 4, Na2C206, Na 2 C05, NaHCO 4 (two isomers) are obtained by the action of gaseous or solid carbon dioxide on the peroxides Na 2 0 2, Na 2 0 3, NaHO 2 (two isomers)in the presence of water at a low temperature (R.Wolffenstein and E.Peltner, Ber., 1908, 41, pp. 275, 280).

This calculus was first applied to the motion of water by d'Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

From a collection of the best experiments by previous workers he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals); and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afforded a simple expression for the velocity of running water.

Bornemann re-examined all these results with great care, and gave formulae expressing the variation of the coefficients of discharge in different conditions (Civil Ingenieur, 1880).

Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855) led him to propose variations in the accepted formulae for the discharge over weirs, and a generation later a very complete investigation of this subject was carried out by H.

Harlacher (1842-1890) contained valuable measurements of this kind, together with a comparison of the experimental results with the formulae of flow that had been proposed up to the date of its publication, and important data were yielded by the gaugings of the Mississippi made for the United States government by A.

The third division would consist of the collections of the so-called Pseudo-leges Canuti, the laws of Edward the Confessor, of Henry I., and the great compilation of the Quadripartitus, then of a number of short notices and extracts like the fragments on the "wedding of a wife," on oaths, on ordeals, on the king's peace, on rural customs (Rectitudines singularum personarum), the treatises on the reeve (gerefa) and on the judge (dema), formulae of oaths, notions as to wergeld, &c. A fourth group might be made of the charters, n as they are based on Old English private and public law and supply us with most important materials in regard to it.

Opinions differ as to the true import of these glosses; some scholars hold that the Salic Law was originally written in the Frankish vernacular, and that these words are remnants of the ancient text, while others regard them as legal formulae such as would be used either by a plaintiff in introducing a suit, or by the judge to denote the exact composition to be pronounced.

These formulae, however, require modification in accordance with the views of Lowry and E.

The identity of the formulae and osazones of d-mannose and d-glucose showed that the stereochemical differences were situated at the carbon atom adjacent to the aldehyde group. Fischer applied a method indicated by Pasteur in converting dextro into laevo-tartaric acid; he found that both d-mannonic and d-gluconic acids (the latter is yielded by glucose on oxidation) were mutually convertible by heating with quinoline under pressure at 140°.

Arabinose being convertible into /-glucose and xylose into l-gulose, the alternative formulae to be considered are CH 2 (OH) - - - +COH CH 2 (OH) + + - COH.

If the asymmetric system adjoining the COH group, which is that introduced in synthesizing the hexose from the pentose, be eliminated, the formulae at disposal for the two pentoses are CH 2 (OH) - - - COH CH 2 (OH)+-- COH.

Hence it follows that the " optical " formulae of the acids derived from two pentoses having the configuration given above will be C02H - 0 - C02H CO 2 H + 0 - C02H, and that consequently only one of the acids will be optically active.

Fischer has proposed formulae for the important disaccharoses, and in conjunction with Armstrong devised a method for determining how the molecule was built up, by forming the osone of the sugar and hydrolysing, whereupon the hexosone obtained indicates the aldose part of the molecule.

The isomerism which occurs as soon as the molecule contains a few carbon atoms renders any classification based on empirical molecular formulae somewhat ineffective; on the other hand, a scheme based on molecular structure would involve more detail than it is here possible to give.

The formulae of the group of substances last mentioned are in harmony with the ordinary views of chemists as to valency, but the formulae NaHg 2, NaCd 2, NaT1 2, AuAl 2 are more surprising.

These have often been employed as empirical formulae (e.g.

In the modified Joule-Thomson equation (17), both c and n have simple theoretical interpretations, and it is possible to express the thermodynamical properties of the substance in terms of them by means of reasonably simple formulae.

Writing formulae (3r) and (33) for the energy and entropy with indeterminate constants A and B, instead of taking them between limits, we obtain the following expressions for the thermodynamic functions in the case of the vapour: " =Solog e 0 - R log e p - ncp/D+A".

The process was usually explained as the result of the action of a spirit, angel or devil, and many unessential formulae, invocations, "calls," written charms with cabbalistic signs, and fumigations, were employed.

The alloys of the formulae AuAg, AuAg 2, AuAg 4 and AuAg 2 o are perfectly homogeneous, and have been studied by Levol.

Less accurate formulae are =p W/(W - W 2), the factor involving the density of the air, and the coefficient of the expansion of the solid being disregarded, and 0 =W/(W - W 1), in which the density of water is taken as unity.

The formulae show the number of cubic centimetres of gas absorbed by i litre of sea-water; t indicates the temperature in degrees centigrade and CI the salinity as shown by the amount of chlorine per mille: 02 = 10.291 - 0 .

In the following table are given the values of the diameters of the molecules of six substances with which it is easy to experiment in the gaseous state, these values being calculated in different ways from formulae supplied by the kinetic theory.

What is given by the formulae is accordingly the mean radius of an irregularly shaped solid (or, more probably, of the region in which the field of force surrounding such a solid is above a certain intensity), and the mean has to be taken in different ways in the different phenomena.

(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.

formulae (5) and (6)] shows that in the normal state the different substances are all at the same temperature.

Merely verbal change in these formulae made without prejudicing the sense does not invalidate the sacrament.

But in that case how can the creed and ritual of baptism, the Lord's Prayer and the Eucharistic formulae, have been kept secret?

It might, therefore, be described as that branch of mathematics which deals with formulae for calculating the numerical measurements of curved lengths, areas and volumes, in terms of numerical data which determine these measurements.

It is also convenient to regard as coming under mensuration the consideration of certain derived magnitudes, such as the moment of a plane figure with regard to a straight line in its plane, the calculation of w]iich involves formulae which are closely related to formulae for determining areas and volumes.

On the other hand, the scope of the subject, as described in § 3, is limited by the nature of the methods employed to obtain formulae which can be applied to actual cases.

Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or geometrical methods.

These investigations lead, in turn, to further formulae, which, though not obtainable by elementary methods, are nevertheless simple in themselves and of practical utility.

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.

This use of formulae for dealing with numbers, which express magnitudes in terms of units, constitutes the broad difference between mensuration and ordinary geometry, which knows nothing of units.

The general formulae applicable to these cases are largely approximative.

As a result of the importance both of the formulae obtained by elementary methods and of those which have involved the previous use of analysis, there is a tendency to dissociate the former, like the latter, from the methods by which they have been obtained, and to regard mensuration as consisting of those mathematical formulae which are concerned with the measurement of geometrical magnitudes (including lengths), or, in a slightly wider sense, as being the art of applying these formulae to specific cases.

Such a body of formulae cannot, of course, be regarded as constituting a science; it has no power of development from within, and can only grow by accretion.

It may be of extreme importance for practical purposes; but its educational value, if it is studied apart from the methods by which the formulae are obtained, is slight.

On the other hand, mensuration, in its practical aspect, is of importance for giving reality to the formulae themselves and to the principles on which they are based.

The commutative law for multiplication is directly illustrated; and subdivisions or groupings of the units lead to such formulae as (a+ a) (b + 0)=.

formulae, not involving the conception of an angle as generated by rotation, belong to this stage; the additional geometrical idea involved being that of the proportionality of the sides of similar triangles.

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

of formulae, will depend on the purpose for which the collection of formulae is required, and may involve the grouping of results obtained by very different methods.

A collection of formulae relating to the circle, for instance, would comprise not only geometrical and trigonometrical formulae, but also approximate formulae, such as Huygens's rule (§ 91), which are the result of advanced analysis.

The present article is not intended to give either a complete course of study or a complete collection of formulae, and therefore such only of the ordinary formulae are given as are required for illustrating certain general principles.

The most important formulae are those which correspond to the use of rectangular Cartesian co-ordinates.

Quadrature-formulae or cubature-formulae may sometimes be conveniently replaced by formulae giving the mean ordinate or mean section.

In the present article the formulae for area or volume will be used throughout.

This is the case, for instance, with formulae which involve 7r or trigonometrical ratios.

(b) It must, however, be remembered that in all practical applications of formulae the data have first to be ascertained by direct or indirect measurement; and this measurement involves a certain margin of error.

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.

It is necessary, in applying formulae to specific cases, not only, on the one hand, to remember that the measurements are only approximate, but also, on the other hand, to give to any ratio such as 7r a value which is at least more accurate than the measurements.

By considering the circle as the limit of a polygon, it follows that the formulae (iii) and (v) of § 26 hold for a right circular cylinder and a right circular cone; i.e.

These formulae also hold for any right cylinder and any cone.

These latter formulae are due to Archimedes.

These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.

- The formulae of § 24 for the area of a trapezoid are of special importance in land-surveying.

We have now to consider the extension of formulae of this kind to other figures, and their application to the calculation of moments and volumes.

Formulae giving the area of a trapezette should in general also be expressed so as to state the value of the mean ordinate (§§ 12 (v), 15, 19).

These cases are important as enabling simpler formulae, involving central differences, to be used (§ 76).

The result of treating this area as if it were the ordinate of a trapezette leads to special formulae, when the data are of the kind mentioned in § 44.

Simpson's two formulae also apply if u is of the form px 3 - }- 5x 2 + rx -}- s.

For functions of higher degrees in x the formulae become more complicated.

The general method of constructing formulae of this kind involves the use of the integral calculus and of the calculus of finite differences.

The general formulae of § 54 (p being replaced (i) by 2m) may in the same way be applied to obtain formulae giving the area of the trapezette in terms of the mid-ordinates of the strips, the series being taken up to b 2f ul m or /th 2J ug m at least, where u is of degree 2f or 2f + I in x.

The formulae can then be applied to finding the moments of certain volumes.

The terms quadratureformula and cubature-formula are sometimes restricted to formulae for expressing the area of a trapezette, or the volume of a briquette, in terms of such data.

The formulae become complicated when the number of strips in each of the minor trapezettes is large.

Some of the formulae obtained by the above methods can be expressed more simply in terms of chordal or tangential areas taken in various ways.

The following are some examples of formulae of this kind, in terms of chordal areas.

There are similar formulae in terms of the tangential areas T1, Thus (iii) of § 68 may be written A -a -1-(9T 1 - T3).

The general method of constructing the formulae of § 7 0 for chordal areas is that, if p, q, r, ...

(iii) Different formulae have to be adopted for different values of m; the method is therefore unsuitable for the construction of a table giving successive values of the area up to successive ordinates.

The above formulae apply only to a parabolic trapezette.

, but also a sufficient number of the ordinates obtained by continuing the series outside the trapezette, at both extremities, we can use central-difference formulae, which are by far the most convenient.

The formulae of § 75 give 11?

The formulae usually employed are A = C1+h) d 3 2 0u0- 02u0+.1.186.z3u0-160p4uo+..

The establishment of these formulae involves the use of the integral calculus.

To find the moments with regard to the central ordinate, we must use this approximate value, and transform by means of the formulae given in § 32.

The formulae can be adapted to the case in which cp(x) is tabulated for xl,.

In order to find the corrections in respect of the terms shown in square brackets in the formulae of § 75, certain ordinates other than those used for C 1 or T I are sometimes found specially.

, we obtain exact formulae corresponding to the approximate formulae of § 70.

The following are the results (for the formulae involving chordal areas), given in terms of differential coefficients and of central differences.

They are not so convenient as the formulae of § 76, but they serve to indicate the degree of accuracy of the approximate formulae.

The application of the methods of §§ 75-79 to calculation of the volume of a briquette leads to complicated formulae.

The formulae of § 82 can be extended to the case of a briquette whose top has close contact with the base all along its boundary; the data being the volumes of the minor briquettes formed by the planes x =x0, x = x i,

and y = yo, y = Yi, The method of constructing the formulae is explained in § 62.

on the assumption that the volume of each minor briquette is concentrated along its mid-ordinate (§ 44), and we then obtain the formulae of correction by multiplying the formulae of § 82 in pairs.

Even where u is an explicit function of x, so that f x udx may be expressed in terms of x, it is often more convenient, for construction of a table of values of such an integral, to use finite-difference formulae.

Each of the above formulae involves an arbitrary constant; but this disappears when we start the additions from 'a known value of X udx.

The formulae may be used for extending the accuracy of tables, in cases where, if v represents the quantity tabulated, hdv/dx or h 2 d 2 v /dx 2 can be conveniently expressed in terms of v and x to a greater degree of accuracy than it could be found from the table.

The use of quadrature-formulae is important in actuarial work, where the fundamental tables are based on experience, and the formulae applying these tables involve the use of the tabulated values and their differences.

The following are instances of the application of approximative formulae to the calculation of the volumes of solids.

Certain approximate formulae for the length of an arc of a circle are obtained by methods similar to those of §§ 71 and 79.

Pierpoint's Mensuration Formulae (1902) is a handy collection.

P. Elderton, Frequency-Curves and Correlation (1906); as to the formulae of § 82, see also Biometrika, v.

Bridge sections designed by this rule differ little from those designed by formulae based directly on Wohler's experiments.

They can be explained, partly by the origin of the State - for the most part through a voluntary union of countries possessed by a strong sense of their own individuality - partly by the influence in Austria of the Germanic spirit, well understood by the Slays, which has nothing of the Latin tendency to reduce all questions of administration to clear-cut formulae as part of a logically consistent system.

As regards its constitution, it has been given at different times the formulae NI 3, NHI 2, NH 2 I, N 2 H 3 I 3, &c., these varying results being due to the impurities in the substance, owing to the different investigators working under unsuitable conditions, and also to the decomposing action of light.

They are a series of dialogues, written first for pupils in the early Paris days as formulae of polite address, but afterwards expanded into lively conversations, in which many of the topics of the day are discussed.

On the contrary, the sacred formulae were revered because they were believed to contain the determination of the highest truths: the knowledge of God and of the mystery of salvation.

But in so doing they did not only repeat the old formulae; the ideas of the men of old sprang into new life.

One special class called kher heb were charged with reciting the divine formulae, which were popularly held to possess magical virtue.

dihydro-compounds, corresponding to the formulae II.

The calculation can be carried out in each region of velocity from the formulae: (25) T(V) - T(v) =k f vvm dv, S(V)-S(v) =k f vvm+ldv I (V)-I(v)=gk v vv m-ldv, and the corresponding integration.

Now calculate the pseudo-velocity uo from =v 95 cos 4) sec n, and then, from the given values of 0 and 8, calculate u e from either of the formulae of (72) or (73): (82) I (u 9) - I (u0) tan 0 - tan 8 C sec n (83) D(ue) =D (uq5) 4)°-B° cos n' Then with the suffix notation to denote the beginning and end of the arc 0-0, mt e = C[Tum) - T (u0)], 5 ((x x9 1l 0.

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.

Two alternative structural formulae have been given to benzoquinone, namely: 1 O-O and 0=(>0 The former, due to C. Graebe (Zeit.

The following are examples of the standing formulae used by the compiler for the purpose :-" In the twentieth year of Jeroboam king of Israel began Asa to reign over Judah.

Students of Tallquist's Maklu series of incantation or of the surpu series edited by Zimmern (in his Beitrage zur Kenntniss der Babylonischen Religion) will recollect the images over which the priest sorcerer recites his formulae.

Thus in arithmetical calculations if the base is not expressed it is understood to be io, so that log m denotes log n m; but in analytical formulae it is understood to be e.

Other formulae which are deducible from this equation are given in the portion of this article relating to the calculation of logarithms.

A very full account of these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol.

In the following account only those formulae and methods will be referred to which would now be used in the calculation of logarithms.

The following still more convenient formulae for the calculation of log e 2, loge3, &c. were given by J.

By means of these formulae Adams calculated the values of log e 2, log e 3, log e s, and loge?

If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus, log i o(I +x) = M(x-2x2+3x3-4x4+&c.), and so on.

von Helmholtz, working on a similar hypothesis, but with a frictional term introduced into his equations, obtained formulae which are applicable to cases of absorption.

- Considerable discussion has taken place as to the structure of the metallic cyanides, since potassium cyanide and silver cyanide react with alkyl iodides to form nitriles and isonitriles respectively, thus apparently pointing to the fact that these two compounds possess the formulae KCN and AgNC. The metallic cyanides are analogous to the alkyl isocyanides, since they form soluble double silver salts, and the fact that ethyl ferrocyanide on distillation yields ethyl isocyanide also points to their isocyanide structure.

Exact formulae are: - Arc =a0, where 8 may be given directly, or indirectly by the relation c=2a sin 28.

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

Exact formulae are: - Area = 2a 2 (0 - sin 0)=1a 2 0 - 4c2 cot zB =Za 2 -2 c?1 (a 2 -4c 2).

Approximate formulae are: - Area = 15 (6c+ 8c 2)h; = 11,/ (c2+h42) h; = i, (7 c +3 a) h, a being the true length of the arc.

We Will Now Show In What Manner This Whole Apparatus Of Methods And Tables May Be Dispensed With, And The Gregorian Calendar Reduced To A Few Simple Formulae Of Easy Computation.

By Means Of The Formulae Which We Have Now Given For The Dominical Letter, The Golden Number And The Epact, Easter Sunday May Be Computed For Any Year After The Reformation, Without The Assistance Of Any Tables Whatever.

LIBER DIURNUS ROMANORUM PONTIFICUM, or "Journal of the Roman Pontiffs," the name given to a collection of formulae used in the papal chancellery in preparing official documents, such as the installation of a pope, the bestowal of the pallium and the grant of papal privileges.

The dip of the horizon and its distance in sea-miles when the height of the observer's eye above the sea-level is h feet, are approximately given by the formulae: Dip=o' 97 -slh; Distance =1 m.

By the time of the VIth Dynasty it was usual to lay the corpse on its left side in the attitude of sleep, and a wooden coffin was often provided upon which were inscribed magic formulae that had already been employed for ages in ritual.

His special interest in Christian Greek was partly the cause of his editing in 1877 The Apologies of Justin Martyr, " which" (to use his own words) "I used unblushingly as a repository for my syntactical formulae."

Miller) collects these formulae.

From these introductions we are able to judge of the style of Pappus's writing, which is excellent and even elegant the moment he is free from the shackles of mathematical formulae and expressions.

crotonic acid, isocrotonic acid and methacrylic acid; the constitutional formulae are HC CO 2 H HC C02H HC CH3 CH3 CH Crotonic Acid.

Schelling himself, as soon as he saw his own formulae exposed in the logic or rather dialectic of his disciple, began to reconsider his philosophy of identity, and brought some powerful objections against both the conclusions and the method of Hegel.

In view of the serious difficulties connected with the inclining of every ship, Smith's formulae for ascertaining and providing for the correction of the heeling error with the ship upright continue to be of great value to safe navigation.

The general formulae of hydroxides are: M i OH, M ii (OH) 2, M il i (OH) 3, M i `'(OH) 4, &c., corresponding to the oxides M21O, M iiO, M21i103 M i °O 2, &c., the Roman index denoting the valency of the element.

The same oxygen ether is formed by the methylation of the silver salt of the normal diazo hydroxide; this points to the conclusion that the isomeric hydroxides, corresponding with the silver derivatives, have the same structural formulae, namely, R N: N OH.

In this case it is readily seen that isomerism introduces itself in the three carbon atom derivative: the propyl alcohols, expressed by the formulae CH 3 CH2 CH 2 0HandCH 3 CHOH CH3, are known as propyl and isopropyl alcohol respectively.

The Leges are divided into Leges nationum Germanicarum, Capitularia regum Francorum, Concilia, Constitutiones imperatorum et regum and Formulae.

Our information about the antrustions is derived from one of the formulae of Marculfus (i.

by means of simple formulae calculate quantities closely agreeing with those obtained from the seismogram.

Of the seven verses of the sura no less than five (verses 1, 2, 3, 4, 6) have an extremely suspicious relationship with the stereotyped formulae of Jewish and Christian liturgies.

On the other hand, the question must remain open whether Mahomet only gave free renderings of the several borrowed formulae, or whether in actually composing them he kept existing models.

The two magic formulae, suras cxiii., cxiv.

and the magical formulae of cxiii., cxiv.

The temple ritual employed in the daily cult is ifiustrated by the scenes depicted on the inner walls of the great temples: the formulae recited during the performance of the ceremonies are recorded at length in the temple of Seti I.

it is often hard to discern much connection between the acts and the formulae recited, but the main thought is clearly that the priest represents Horus, the pious son of the dead divinity Osiris.

a late book has preserved to us the magical formulae that were repeated by the wise kher-heb priest (who in the necropolis performed the functions of taricheutes, embalmer), as each bandage was applied.

Symbolical or imitative acts, accompanied by spoken formulae of set form and obscure content, accomplished, by some peculiar virtues of their own, results that were beyond the power of human hands and brain..

It is extremely hard to draw any fixed line in Egypt between magic and medicine; but it is curious to note that simple diagnoses and prescriptions were employed for the more curable diseases, while magical formulae and amulets are reserved for those that are harder to cope with, such as the bites of snakes and the stings of scorpions.

The formulae recited for such purposes are not purely cabalistic, though inasmuch as mystery is of the very essence of magic, foreign words and outlandish names occur in them by preference.

When a pretence was made of opening, with an iron instrument, the mouth of the divine statue, to the accompaniment of recited formulae, this can hardly be termed anything but magic. Similarly, the potency attributed to ushebli-figures and the copies of the Book of the Dead deposited in the tombs is magical in quality.

Such formulae are found in the ancient Greek liturgies, e.g.

Usually this lime is regarded only as mechanically mixed with the bleaching-compound, CaOCl 2, but some chemists adopt formulae in which this lime is equally represented.

These compounds may be represented by the following formulae: OH OH OH OH / / CO /OH HO/ / CO /OH / / CO /OH /CO/OH CO/?/ Unco/L) Ho?Co?%?% `Co/%Oh.

Partly in consequence of this mystery and partly in accord with widespread superstitions, the Tetragrammaton figures in magical formulae from the time of the Gnostics, and on amulets.

At the same time, since the invoking of the divine powers was the essential element in the incantations, in order to make the magic formulae as effective as possible, a large number of the old local deities are introduced to add their power to the chief ones; and it is here that the astral system comes into play through the introduction of names of stars, as well as through assigning attributes to the gods which clearly reflect the conception that they have their seats in the heavens.

The ritual alone which accompanied divination practices and incantation formulae and was a chief factor in the celebration of festival days and of days set aside for one reason or the other to the worship of some god or goddess or group of deities, is free from traces of the astral theology.

Comparatively Simple Formulae, Therefore, Suffice For Its Expression To 1 In 10,000, Which Is Beyond The Limits Of Accuracy Of The Observations.

It Is More Convenient In Practice To Use A Few Simple Formulae, Than To Attempt To Represent The Whole Range By A Single Complicated Expression Below 20° C. S=0.9982 O 000,0045 (T 40) 2 O.

The following table of values, calculated from these formulae, is taken from the Brit.

The Value 4.180 Joules At 20° C. Is The Mean Between Rowland'S Corrected Result 4.181 And The Value 4.179, Deduced From The Experiments Of Reynolds And Moorby On The Assumption That The Ratio Of The Mean Specific Heat O° To 100° To That At 20° Is 1.043'6, As Given By The Formulae Representing The Results Of Callendar And Barnes.

The Mean Specific Heat, Over Any Range Of Temperature, May Be Obtained By Integrating The Formulae Between The Limits Required, Or By Taking The Difference Of The Corresponding Values Of The Total Heat H, And Dividing By The Range Of Temperature.

It May Be Remarked That Starting From The Same Value At 5°, For The Sake Of Comparison, Rowland'S Values Of The Total Heat Agree To I In 5000 With Those Calculated From The Formulae.

In Any Case It May Be Remarked That Formulae Such As Those Of Jamin, Henrichsen, Baumgartner, Winkelmann Or Dieterici, Which Give Far More Rapid Rates Of Increase Than That Of Regnault, Cannot Possibly Be Reconciled With His Observations, Or With Those Of Reynolds And Moorby, Or Callendar And Barnes, And Are Certainly Inapplicable Above Loo° C.

The Following Formulae Represent His Results For The Specific Heat S At Constant Volume In Terms Of The Density D In Gms. Per C. C.: Air, S = 0.1715 0 028D, C02, S = 0.165 0.213D } O 34D2.

Enough has been said to justify the great place assigned to Aristotle in the history of logic. Without pressing metaphysical formulae in logic proper, he analysed formal;implica tion, grounded implication as a mode of knowledge Summary.

Particulars to controlling formulae.

But consider Bacon's own doctrine of forms. Or watch the mathematical physicist with his formulae.

They are formulae for the control of the activities and the production of the qualities of bodies.

The conclusions seem not merely to fall within, but to depend on these organic and controlling formulae.

He prescribed regulative or limiting formulae for research as it was actually conducted in his world.

1908) that, if the accurate formulae for the vapour-pressure given below are employed, the results for strong solutions are consistent with a very slight, but important, modification of Raoult's law.

(to) He obtained similar formulae for other vapours, but the experiments were not so complete or satisfactory as in the case of steam, which may conveniently be taken as a typical vapour in comparing theory and experiment.

S = 475, greatly increased the apparent discrepancy between Regnault's and Rankine's formulae for the total heat.

Values calculated from these formulae are given in the table below.

Regnault's formula for the total heat is here again seen to be inadmissible, as it would make the latent heat of steam vanish at about 870° C. instead of at 365° C. It should be observed, however, that the assumptions made in deducing the above formulae apply only for moderate pressures, and that the formulae cannot be employed up to the critical point owing to the uncertainty of the variation of the specific heats and the cooling effect Q at high pressures beyond the experimental range.

Many attempts have been made to construct formulae representing the deviations of vapours from the ideal state up to the critical point.

The values of the saturation-pressure have been ver y accurately determined for the majority of stable substances, and a large number of empirical formulae have been proposed to represent the relation between pressure and temperature.

These formulae are important on account of the labour and ingenuity expended in devising the most suitable types, and also as a convenient means of recording the experimental data.

In the following list, which contains a few typical examples, the different formulae are arranged to give the logarithm of the saturation-pressure p in terms of the absolute temperature 0.

As originally proposed, many of these formulae were cast in exponential form, but the adoption of the logarithmic method of expression throughout the list serves to show more clearly the relationship between the various types.

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