# How to use *Formulae* in a sentence

Traditionalists would prefer to follow

**formulae**that they know work, rather than try new methods.In the second of these notices he gives the general

**formulae**without demonstrations.Rational

**formulae**of a much simpler description than these graphic**formulae**are generally employed.You will be able to work more efficiently if you just memorize the

**formulae**and don't have to keep looking them up.Before explaining these

**formulae**it will be necessary, however, to consider the differences in combining power exhibited by the various elements.AdvertisementIt is important to memorize all the chemical

**formulae**for the reaction we are looking at in today's lab.He calculated the answer to the math problems using the

**formulae**provided by the teacher.The baker did an excellent job of creating

**formulae**for making her cakes perfectly moist and delicious.Lowry and Armstrong represent these compounds by the following spatial

**formulae**which postulate a y-oxidic structure, and 5 asymmetric carbon atoms, i.e.In conduct they shrank from

**formulae**.AdvertisementThese

**formulae**are supported by many considerations, especially by the selective CH 2 OH CH20H CH OH CH OH C C H OThe 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.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.AdvertisementIt 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.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.What is more important is to verify particular examples of general

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

**formulae**.AdvertisementThey 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.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.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.AdvertisementThe 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.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).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.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.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**.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.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.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.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)=.This involves the use of Cartesian co-ordinates, and leads to important general

**formulae**, such as Simpson's formula.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.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.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.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 above

**formulae**apply only to a parabolic trapezette.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.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,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.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.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.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.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.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.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.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**."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.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 Leges are divided into Leges nationum Germanicarum, Capitularia regum Francorum, Concilia, Constitutiones imperatorum et regum and

**Formulae**.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.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.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.The following table of values, calculated from these

**formulae**, is taken from the Brit.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.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.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.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.If we assume

**formulae**of the simple type A+B/0 for two different substances which have the same vapour-pressure p at the absolute temperatures 0' and 0" respectively, we may write log p=A'+B'/0'= A"+B"/0", .The

**formulae**of Young and Roche are purely empirical, but give very fair results over a wide range.The

**formulae**of Rankine and Unwin, though probably less accurate over the whole range, are much simpler and more convenient in practice than that of Biot, and give results which suffice in accuracy for the majority of purposes.The empirical

**formulae**above quoted must be compared and tested in the light of the theoretical relation between the latent heat and the rate of increase of the vapour-pressure (dp/d0), which is given by the second law of thermodynamics, viz.Taking for ice and water the following numerical data, L = 674.7, 6 74.7, L 1 =595.2, L r = 79.5, R = o 11 03 cal./deg., po = 4.61 mm., s-S = 519 cal./deg., and assuming the specific heat of ice to be equal to that of steam at constant pressure (which is sufficiently approximate, since the term involving the difference of the specific heats is very small), we obtain the following numerical

**formulae**, by substitution in (23), Ice..The values of the saturation-pressure given in the last column are calculated by formula (25), which agrees with Regnault's observations better than his own empirical

**formulae**.Although it is not possible to represent the properties of steam in this manner up to the critical temperature, the above method appears more satisfactory than the adoption of the inconsistent and purely empirical

**formulae**which form the basis of most tables at the present time.An immense mass of material has been collected on the subject of vapour-pressures and densities, the greater part of which will be found in Winkelmann's Handbook, in Landolt's and Bornstein's Tables, and in similar compendiums. The results vary greatly in accuracy, and are frequently vitiated by errors of temperature measurement, by chemical impurities and surface condensation, or by peculiarities of the empirical

**formulae**employed in smoothing the observations; but it would not be within the scope of the present article to discuss these details.Oxyfluorides of the

**formulae**WOF 4 and WO 2 F 2 are also known.These

**formulae**give a means of constructing the resultant by means of any transversal AB cutting the lines of action.As an example of the use of the

**formulae**we may determine the maximum span for a wire of given material.From the purely kinematic point of view, the t of our

**formulae**may be any continuous independent variable, suggested (it may be) by some physical process.The preceding

**formulae**are sufficient for the, treatment of instantaneous impulses.The n

**formulae**of this type represent a normal mode of free vibration; the individual particles revolve as a rule in elliptic orbits which gradually contract according to the law indicated by the exponential factor.The only modification required in the

**formulae**is, that in equation (26) the difference of the angular velocities should be substituted for their sum.The length L of an endless belt connecting a pair of pulleys whose effective radii are r,, r,, with parallel axes whose distance apart is c, is given by the following

**formulae**, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.The interesting and important part of the investigation is that a number of experiments were made on small shafts arranged in different ways and loaded in different ways, and the speed at which whirling actually occurred was compared with the speed calculated from

**formulae**of the general type indicated above.A special feature of the Sakti cult is the use of obscure Vedic mantras, often changed so as to be quite meaningless and on that very account deemed the more efficacious for the acquisition of superhuman powers; as well as of mystic letters and syllables called bija (germ), of magic circles (chakra) and diagrams (yantra), and of amulets of various materials inscribed with

**formulae**of fancied mysterious import.For a generation nursed in decadent scholasticism and stereotyped theological

**formulae**it was the fountain of renascent youth, beauty and freedom, the shape in which the Helen of art and poetry appeared to the ravished eyes of medieval Faustus.The settler who went into the woods might know neither the name nor the sex of the indwelling numen; " si deus si dea," " sive mas sive femina," ran the old

**formulae**.Such ritual use of oil as a o payls or seal may have been suggested in old religions by the practice of keeping wine fresh in jars and amphorae by pouring on a top layer of oil; for the spoiling of wine was attributed to the action of demons of corruption, against whom many ancient

**formulae**of aversion or exorcism still exist.The aberrations can also be expressed by means of the "characteristic function " of the system and its differential coefficients, instead of by the radii, &c., of the lenses; these

**formulae**are not immediately applicable, but give, however, the relation between the number of aberrations and the order.These propositions may be derived from the

**formulae**given above, or proved directly by purely geometrical methods.That these alkaloids are closely related may be suspected from their empirical

**formulae**, viz.morphine = C17H19N03, codeine = C18H21N03, thebaine = C19H21N03.**Formulae**have been proposed by Pschorr and Knorr explaining this and other decompositions (in Pschorr's formula the morphine ring system is a fusion of a phenanthrene and pyridine nucleus); another formula, containing a fusion of a phenanthrene with a pyrrol ring, was proposed by Bucherer in'1907.Sympathy and symbolism, association of ideas and analogy, together with a certain amount of observation, are the explanation of the great mass of heteroscopic divinatory

**formulae**.We do not attempt to fit them to absolute

**formulae**, but continually adapt them to a changing environment.It substituted the work of the genius for the miraculous intervention of Providence, but, apart from certain abstract

**formulae**such as Truth and Right, knew nothing of why or how.It may reasonably be supposed, not only that they constructed the external framework of many chapters, and also made some additions of their own - a necessary process in order to weld their motley collection of fragments into a new and coherent book - but also that they fabricated anew many

**formulae**and imitative passages on the model of the materials at their disposal.The Yasna and many Yashts in great part consist of

**formulae**of prayer which are as poor in contents as they are rich in verbiage.The results thus derived are, however, based on imperfect

**formulae**.His philosophical treatises abound with incoherent

**formulae**to which, according to their inventor, every demonstration in every science may be reduced, and posterity has ratified Bacon's disdainful verdict on Lull's pretensions as a thinker; still the fact that he broke away from the scholastic system has recommended him to the historians of philosophy, and the subtle ingenuity of his dialectic has compelled the admiration of men so far apart in opinion as Giordano Bruno and Leibniz.The sectional areas are computed by the use of the ordinary

**formulae**for columns and struts.It is usual in both girders and beams to provide not only for the safe support of the greatest possible distributed load, but for the greatest weight, such as that of a safe or other heavy piece of furniture which may be moved over the floor at its weakest points, the centres of the girders and beams. It must always be borne in mind that the

**formulae**for the ultimate strength of the " I " beams only hold good when the upper chord or flange is supported laterally.They speak of the ordination of bishops (the so-called Clementine Liturgy is that which is directed to be used at the consecration of a bishop, cc. 5-15), of presbyters, deacons, deaconesses, subdeacons and lectors, and then pass on to confessors, virgins, widows and exorcists; after which follows a series of canons on various subjects, and liturgical

**formulae**.His Elements of Geometry first appeared in 1795 and have passed through many editions; his Outlines of Natural Philosophy (2 vols., 1812-1816) consist of the propositions and

**formulae**which were the basis of his class lectures.More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical

**formulae**.But to the younger generation the irreconcilability of modern scholarship and medieval

**formulae**of faith became more and more evident.The influence of literature on Burke lay partly in the direction of emancipation from the mechanical

**formulae**of practical politics; partly in the association which it engendered, in a powerful understanding like his, between politics and the moral forces of the world, and between political maxims and the old and great sentences of morals; partly in drawing him, even when resting his case on prudence and expediency, to appeal to the widest and highest sympathies; partly, and more than all, in opening his thoughts to the many conditions, possibilities and "varieties of untried being," in human character and situation, and so giving an incomparable flexibility to his methods of political approach.The most prolonged mathematical reasoning, and the most intricate

**formulae**, were given with almost infallible accuracy from the resources of his extraordinary memory.Reddish brown amorphous powders of the

**formulae**2FeC1 3NO and 4FeC13NO are obtained by passing the gas over anhydrous ferric chloride.Such foresight had its reward, the more because it was buttressed during the debates of the Convention by the same readiness in debate, the same clear recognition of essentials, the same natural disposition towards compromise on details, and the same quickness in producing verbal

**formulae**, as Smuts had already shown in the Transvaal Parliament.They merely analysed conventional

**formulae**, much in the manner of certain modern so-called " scientific " moralists.Improved practice, again, reacted upon theory by bringing to notice residual errors, demanding the correction of

**formulae**, or intimating neglected disturbances.The relations of the quantities involved may be shown by simple trigonometric

**formulae**.It is much to be regretted that the Atharva-Veda, which contains the magical

**formulae**and incantations of the Vedic Indians, is still untranslated, though, by the very nature of its theme, it must contain matter of extreme antiquity and interest.The

**formulae**here summarized, with others, are familiar in the marchen of Samoyeds, Zulus, Bushmen, Hottentots and Red Indians.Still, the logic of Albertus Magnus and succeeding doctors was largely indebted to him for its

**formulae**.Fresnel obtained his

**formulae**by assuming that the optical difference of media is due to a change in the effective density of the ether, the elasticity being the same - an assumption inconsistent with his theory of double refraction - and was led to the result that the vibrations are perpendicular to the plane of polarization.Franz Neumann and James MacCullagh, starting from the opposite assumption of constant density and different elasticities, arrived at the same

**formulae**for the intensities of the reflected light polarized in the principal azimuths, but in this case the vibrations must be regarded as parallel to the plane of polarization.Now Fresnel's

**formulae**were obtained by assuming that the incident, reflected and refracted vibrations are in the same or opposite phases at the interface of the media, and since there is no real factor that converts cos T into cos (T+p), he inferred that the occurrence of imaginary expressions for the coefficients of vibration denotes a change of phase other than 7r, this being represented by a change of sign.Applying this interpretation to the

**formulae**given above, it follows that when the incident light is polarized at an azimuth a to the plane of incidence and the second medium is the less refracting, the reflected light at angles of incidence exceeding the critical angle is elliptically polarized with a difference of phase A between the components polarized in the principal azimuths that is given by tan (A/2) =cot i l l (1 - µ 2 cosec 2 i).It was at first supposed that the defect of Fresnel's

**formulae**was due to the neglect of the superficial undulations that, on a rigorous elastic solid theory of the ether, are called into existence at reflection and refraction.The effect of such a transition-layer can easily be calculated, at least approximately; but it is of little use to take account of it except in the case of a theory of reflection that gives Fresnel's

**formulae**as the result of an abrupt transition.**Formulae**for metallic reflection may be obtained from Fresnel's expressions by writing the ratio sin i / sin r equal to a complex quantity, and interpreting the imaginary coefficients in the manner explained above.The following

**formulae**are some of those employed for this purpose by different observers E t = bt +ct 2..These

**formulae**are not so simple and convenient as Tait's, though apparently founded on a more simple assumption, but they frequently represent the observations more closely.Many other

**formulae**have been suggested.They cannot be changed into one another, and have been given the

**formulae**C 6 H 4 (CC1 3) (COC1) and C6H4(CC12)20.For the graves yielded not only new types of statues, bronzes, ivory carvings and painted pottery - all of the highest artistic value - but also a large number of stone stelae inscribed with funerary

**formulae**in the Meroitic script.The work left by Napier is, however, rough and unfinished, and it is uncertain whether he knew of the other

**formulae**or not.An analysis of the

**formulae**contained in the Descriptio and Constructio is given by Delambre in vol.He did not, however, confine himself to the consideration of forms involving only three indeterminates, but succeeded in establishing the principles on which the extension to the general case of n indeterminates depends, and obtained the general

**formulae**, thus effecting what is probably the greatest advance made in the subject since the publication of Gauss's Disquisitiones arithmeticae.As corollaries to the general

**formulae**he adds the**formulae**relating to the representation of a number as a sum of five squares and also of seven squares.The fact that a question of which Smith had given the solution in 1867, as a corollary from general

**formulae**governing the whole class of investigations to which it belonged, should have been set by the Academie as the subject of their great prize shows how far in advance of his contemporaries his early researches had carried him.One of his earliest publications gave the complete theory and

**formulae**of the phenomenon known as Brownian motion, which had puzzled physicists for nearly 80 years.The

**formulae**of Kekule, Divers and Armstrong have been discarded, and it remains to be shown whether Nef's carbonyloxime formula (or the bimolecular formula of Steiner) or Scholl's glyoxime peroxide formula is correct.Thus the most prominent subdivision of the older school, the Taittiriyas, in their Samhita, have treated the main portion of the ceremonial in this promiscuous fashion, and to add to the confusion they have, by way of supplement, put forth a so-called Taittiriya-brahmana, which, so far from being a real Brahmana, merely deals with some additional rites in the same confused mixture of sacrificial

**formulae**and dogmatic explanations.The former contained 55, or, according to some

**formulae**, 72 ingredients, and occurs in all the dispensatories, from that of Corvus Valerius up to the pharmacopoeias of the 19th century; and aromatic preparations of opium are still used, under the name of Theriaka in Persia.The Pharmaceutical Society on the other hand has also published a Pharmaceutical Journal Formulary, including several hundred

**formulae**of proprietary medicines sold by pharmacists, so that it is now possible for any medical man to ascertain what they contain.The government accepts all the therein published

**formulae**as " known, admitted and approved " remedies, and therefore not requiring a patent medicine stamp. In this way widely advertised secret remedies can be replaced by medicines of known composition and accepted value in any part of the world.To these also belong the rite of wµoc¢ayta,and the communication of liturgical

**formulae**for the guidance of the soul of the dead man on his way to the underworld, which also served as credentials to the gods below.By combining the four symbols A, B, X, Y, according to their presence or absence, Garrod got a considerable number of

**formulae**, each of which was overruled, so to speak, by the two categories of the presence or absence of the ambiens muscle.Boetius himself tells us in his preface addressed to his father-inlaw Symmachus that he had taken liberties with the text of Nicomachus, that he had abridged the work when necessary, and that he had introduced

**formulae**and diagrams of his own where he thought them useful for bringing out the meaning.This paper of Thomson's, whose ideas Maxwell afterwards developed in an extraordinary manner, seems to have given the first hint that there are at least two perfectly distinct methods of arriving at the known

**formulae**of statical electricity.Berthelot's notation defines both initial and final systems by giving the chemical equation for the reaction considered, the thermal effect being appended, and the state of the various substances being affixed to their

**formulae**after brackets.Ostwald has made the further proposal that the

**formulae**of solids should be printed in heavy type (or within square brackets), of liquids (solutions, &c.) in ordinary type, and of gases in italics (or within curved brackets), so that the physical state of the substances 'might be indicated by the equation itself.It is in fact one of the peculiarities of this theology, which professed to be at once churchly and philosophical, that most of its

**formulae**could be interpreted and appreciated in utramque partem.It seems to have been common among the Jews, and the case of the witch of Endor is narrated in a way to suggest something beyond fraud; in the book of magic which bears the name of Dr Faustus may be found many of the

**formulae**for raising demons; in England may be mentioned especially Dr Dee as one of the most famous of those who claimed before the days of modern spiritualism to have intercourse with the unseen world and to summon demons at his will.In the development of the atomic theory and the deduction of the atomic weights of elements and the

**formulae**of compounds, Dalton's arbitrary rules failed to find complete acceptance.For example, at first he represented ferrous and ferric oxides by the

**formulae**Fe02, Fe03, and by the analogy of zinc and other basic oxides he regarded these substances as constituted similarly to Fe02, and the acidic oxides alumina and chromium oxide as similar to FeO 3.Laurent generally agreed, except when the theory compelled the adoption of

**formulae**containing fractions of atoms; in such cases he regarded the molecular weight as the weight occupying a volume equal to four unit weights of hydrogen.Kekule was the forerunner of his celebrated benzene theory in particular, and of the universal application of structural

**formulae**to the representation of the most complex organic compounds equally lucidly as the representation of the simplest salts.Wislicenus, van't Hoff and others showed that substances having the same graphic

**formulae**vary in properties and reactions, and consequently the**formulae**need modification in order to exhibit these differences.In all cases it is usual to represent substances by

**formulae**which to the best of our knowledge express their molecular composition in the state of gas, and not merely the relative number of atoms which they contain; thus, acetic acid consists of carbon, hydrogen and oxygen in the proportion of one atom of carbon, two of hydrogen, and one of oxygen, but its molecular weight corresponds to the formula C211402, which therefore is always employed to represent acetic acid.When chemical change is expressed with the aid of molecular

**formulae**not only is the distribution of weight represented, but by the mere inspection of the symbols it is possible to deduce from the law of gaseous combination mentioned above, the relative volumes which the agents and resultants occupy in the state of gas if measured at the same temperature and under the same pressure.**Formulae**which merely express the relative number of atoms of the different elements present in a compound are termed empirical**formulae**, and the**formulae**of all compounds whose molecular weights are undetermined are necessarily empirical.In addition to empirical and molecular

**formulae**, chemists are in the habit of employing various kinds of rational**formulae**, called structural, constitutional or graphic**formulae**, &c., which not only express the molecular composition of the compounds to which they apply, but also embody certain assumptions as to the manner in which the constituent atoms are arranged, and convey more or less information with regard to the nature of the compound itself, viz.It is not to be supposed that there are any actual bonds of union between the atoms; graphic

**formulae**such as these merely express the hypothesis that certain of the atoms in a compound come directly within the sphere of attraction of certain other atoms, and only indirectly within the sphere of attraction of others, - an hypothesis to which chemists are led by observing that it is often possible to separate a group of elements from a compound, and to displace it by other elements or groups of elements.Constitutional

**formulae**like these, in fact, are nothing more than symbolic expressions of the character of the compounds which they represent, the arrangement of symbols in a certain definite manner being understood to convey certain information with regard to the compounds represented.Constitutional

**formulae**become of preponderating importance when we consider the more complicated inorganic and especially organic compounds.By continuing the introduction of methyl groups we obtain three series of homologous hydro carbons given by the general

**formulae**CnH2,i 4.2, CnH2n, and CnH2n 2, each member differing from the preceding one of the same series by CH 2.The ring of this compound is ruptured by caustic soda with the formation of perchlorvinyl acrylic acid (5), which gives on reduction ethidine propionic acid (6), a compound containing five of the carbon atoms originally in the benzene ring (see Zincke, Ber., 18 94, 27, p. 33 6 4) (the carbon atoms are omitted in some of the

**formulae**).He admitted the possibility of the

**formulae**of Kekule, Claus, Dewar and Ladenburg, although as to the last di-trimethylene derivatives should be possible reduction products, being formed by severing two of the prism edges; and he attempted to solve the problem by a systematic investigation of the reduced phthalic acids.For succinosuccinic ester, formed by the action of sodium on two molecules of succinic ester, has either of the

**formulae**(I) or (II); oxidation of the free acid gives dioxyterephthalic acid in which the para-positions must remain substituted as in (I) and (II).The relative merits of the

**formulae**of Kekule, Claus and Dewar were next investigated by means of the reduction products of benzene, it being Baeyer's intention to detect whether double linkages were or were not present in the benzene complex.Wislicenus, Le Bel and van 't Hoff has resulted in the introduction of another condition which

**formulae**for the benzene complex must satisfy, viz.In order to adapt this formula to logarithms, we introduce a subsidiary angle p, such that cot p = cot l cos t; we then have cos D = sin 1 cos( - p) I sin p. In the above

**formulae**our earth is assumed to be a sphere, but when calculating and reducing to the sea-level, a base-line, or the side of a primary triangulation, account must be taken of the spheroidal shape of the earth and of the elevation above the sealevel.In the

**formulae**of blessings that follow, the special efficacy against devils is implied by the aspersion with holy water; the benedictions themselves are usually merely invocative of the divine protection or assistance, though, e.g., in the form for blessing sick animals the priest prays that "all diabolic power in them may be destroyed, and that they may be ill no longer."At the same time, the essence of eclecticism is the refusal to follow blindly one set of

**formulae**and conventions, coupled with a determination to recognize and select from all sources those elements which are good or true in the abstract, or in practical affairs most useful ad hoc. Theoretically, therefore, eclecticism is a perfectly sound method, and the contemptuous significance which the word has acquired is due partly to the fact that many eclectics have been intellectual trimmers, sceptics or dilettanti, and partly to mere partisanship. On the other hand, eclecticism in the sphere of abstract thought is open to this main objection that, in so far as every philosophic system is, at least in theory, an integral whole, the combination of principles from hostile theories must result in an incoherent patchwork.Resuming the inquiry into the invariability of mean motions, Poisson carried the approximation, with Lagrange's

**formulae**, as far as the squares of the disturbing forces, hitherto neglected, with the same result as to the stability of the system.He proposed to apply the same principles to the calculation of the disturbances produced in the rotation of the planets by external action on their equatorial protuberances, but was anticipated by Poisson, who gave

**formulae**for the variation of the elements of rotation strictly corresponding with those found by Lagrange for the variation of the elements of revolution.The Schoolmen contemplate the universe of nature and man not with their own eyes but in the glass of Aristotelian

**formulae**.Miscellaneous Developments in Arithmetic. - The following are matters which really belong to arithmetic; they are usually placed under algebra, since the general

**formulae**involve the use of letters.An alternative method, which is in some ways preferable, is to complete the table of differences by repeating the differences of the highest order that will be taken into account (see Interpolation), and then to use central-difference

**formulae**.There is a tendency to eclectic views embracing the more attractive features of the various theories; and attempts are made to adapt, interpret and qualify the imagery and language of older

**formulae**, in order so to speak, to issue them afresh in new editions, compatible with modern natural science, psychology and historical criticism.Euphratean exploration has so far brought to light no traces of ecliptical partition by the moon's diurnal motion, unless, indeed, zodiacal associations be claimed for a set of twenty-eight deprecatory

**formulae**against evil spirits inscribed on a Ninevite tablet.4 The safest general conclusions regarding this disputed subject appear to be that the sieu, distinctively and unvaryingly Chinese, cannot properly be described as divisions of a lunar zodiac, that the nakshatras, though of purely Indian origin, became modified by the successive adoption of Greek and Chinese rectifications and supposed improvements; while the manazil constituted a frankly eclectic system, in which elements from all quarters were combined.**Formulae**of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.In these cases the curvature of the trajectory becomes considerable, and the

**formulae**employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.He was sincerely religious; but his wellmeant efforts to unite the Lutheran and Reformed Churches, in celebration of the tercentenary of the Reformation (1817), revealed the limits of his paternal power; eleven years passed in vain attempts to devise common

**formulae**; a stubborn Lutheran minority had to be coerced by military force, the confiscation of their churches and the imprisonment or exile of their pastors; not till 1834 was outward union secured on the basis of common worship but separate symbols, the opponents of the measure being forbidden to form communities of their own.From Poisson's equations Archibald Smith deduced the

**formulae**given in the Admiralty Manual for Deviations of the Compass (1st ed., 1862), a work which has formed the basis of numerous other manuals since published in Great Britain and other countries.In the Saite period a sort of standard edition was drawn up, consisting of 165 chapters in a fixed order and with a common title the book of going forth in the day; this recension was published by Lepsius in 1842 from a Turin papyrus Like the Pyramid texts, the Book of the Dead served a funerary purpose, but its contents are far more heterogeneous; besides chapters enabling the dead man to assume what shape he will, or to issue triumphant from the last judgment, there are lists of gates to be passed and demons to be encountered in the nether world,

**formulae**such as are inscribed on sepulchral figures and amulets, and even hymns to the sun-god.In these series we can trace the attempt to gather the incantation

**formulae**and prayers produced in different centres, and to make them conform to the tendency to centralize the cult in the worship of Marduk and his consort in the south, and of Assur and Ishtar in the north.The value of these works is impaired somewhat by Baur's habit of making the history of dogma conform to the

**formulae**of Hegel's philosophy, a procedure "which only served to obscure the truth and profundity of his conception of history as a true development of the human mind" (Pfleiderer).Secondly, it does not content itself with the mere

**formulae**of thinking, but pushes forward to theories of method, knowledge and science; and it is a hopeful sign to find this epistemological spirit, to which England was accustomed by Mill, animating German logicians such as Lotze, Daring, Schuppe, Sigwart and Wundt.Plucker communicated his

**formulae**in the first place to Crelle's Journal (1834), vol.In his researches, Roscoe showed that the atomic weight of the metal as determined by Berzelius and the

**formulae**given to the oxides were incorrect, and pointed out that the element falls into its natural place in group V of the periodic classification along with phosphorus and arsenic, and not in the chromium group where it had originally been placed.Of these

**formulae**'(chosen because illustrated by Greek heroic legends) - (I) is a sanction of barbarous nuptial etiquette; (2) is an obvious ordinary incident; (3) is moral, and both (3) and (1) may pair off with all the myths of the origin of death from the infringement of a taboo or sacred command; (4) would naturally occur wherever, as on the West Coast of Africa, human victims have been offered to sharks or other beasts; (5) the story of flight from a horrible crime, occurs in some stellar myths, and is an easy and natural invention; (6) flight from wizard father or husband, is found in Bushman and Namaqua myth, where the husband is an elephant; (7) success of youngest brother, may have been an explanation and sanction of " tungsten-recht " - Maui in New Zealand is an example, and Herodotus found the story among the Scythians; (8) the bride given to successful adventurer, is consonant with heroic manners as late as Homer; (9) is no less consonant with the belief that beasts have human sentiments and supernatural powers; (to) the " strong man," is found among Eskimo and Zulus, and was an obvious invention when strength was the most admired of qualities; (II) the baffled ogre, is found among Basques and Irish, and turns on a form of punning which inspires an " ananzi " story in West Africa; (12) descent into Hades, is the natural result of the savage conception of Hades, and the tale is told of actual living people in the Solomon Islands and in New Caledonia; Eskimo Angekoks can and do descend into Hades - it is the prerogative of the necromantic magician; (13) " the false bride," found among the Zulus, does not permit of such easy explanation - naturally, in Zululand, the false bride is an animal; (14) the bride accused of bearing be 1st-children, has already been disposed of; the belief is inevitable where no distinction worth mentioning is taken between men and animals.They recognize that elements and compounds can be represented by symbols and

**formulae**.Like the semantics of modal logic, the semantics of relevance logic relativises truth of

**formulae**to worlds.Moreover, some students skate by mathematically merely by memorizing the

**formulae**required for problem solving.Students who merely memorize

**formulae**repeatedly will not do well in higher levels of math that require critical thinking abilities to maximize problem solving.When it comes to Scentsy combining scents and recipes, even where exact

**formulae**aren't spelled out for you, you can easily utilize your own creativity alongside general guidelines to create customized scents for your home.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**.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**.These

**formulae**, however, only apply to aliphatic amines; the results obtained in the aromatic series are in accordance with the usual**formulae**.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.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.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.Chandler has constructed empirical

**formulae**representing the motion, with the results already given, while Albrecht has determined the motion of the pole from observation simply, without trying to represent it either by a formula or by theory.It is noteworthy that the difference between Albrecht's numerical results and Chandler's

**formulae**is generally less than o 05".The fluctuations derived from this international work during the last seven years deviate but slightly from Chandler's

**formulae**though they show a markedly smaller value of the annual term.The series will thus possess the following general

**formulae**M 2 CrO 4 M2Cr207 M2Cr30,0 &c. (M =one atom of a normal chromate bichromate trichromate monovalent metal.) Chromates.As a convenient method of expressing the arrangement of the parts of the flower, floral

**formulae**have been devised.However, in common with many

**formulae**there is an important corollary.Extracting relationships from data, manipulation, solving simple equations, rearranging

**formulae**.I have now applied similar spreadsheet

**formulae**to Pensions.It noted that suppliers of soya protein isolate, used in the manufacture of soya-based

**formulae**, had investigated means of reducing phytoestrogen levels.The site also provides the mathematical

**formulae**for the units given.You will get complexes formed involving hydroxide ions, but the

**formulae**of these aren't very clear-cut.Based on these three analytic inverses, three

**formulae**were developed for computing the inverse numerically.Drawing structural

**formulae**for the last pair of models gives two possible isomers.These

**formulae**will also give settings for the assembly jig.Propositional logic Define a data type to represent

**formulae**of propositional logic.Secrecy concerning practical

**formulae**of ceremonial magic is also advisable, for if they are used indiscriminately, the virtue goes out of them.Structural isomerism occurs when two or more organic compounds have the same molecular

**formulae**, but different structures.This does not apply to

**formulae**got by rote, which are stages on the road to nowhere but second childhood and the grave.Like the semantics of modal logic, the semantics of modal logic, the semantics of relevance logic relativises truth of

**formulae**to worlds.He was too shrewd a student of his fellow- men to remain long content with rigid

**formulae**of conduct.There are many

**formulae**relating the sides and angles of a spherical triangle.Relationships and common

**formulae**from elementary trigonometry, including the basic properties of sine and cosine.Its visible and religious forms then give way to abstract

**formulae**, which in their turn are slowly replaced by the rational manifestation of the philosophic principles of law that gains the victory in the final stage of development, designated by Vico as that of civil and human law.The purins themselves may be considered as a combination of the pyrimidine and glyoxaline ring systems. For

**formulae**see below; the numbers about the first ring explain the orientation of pyrimidine derivatives.Though rejected by the Jesuits, who found peripatetic

**formulae**a faithful weapon against the enemies of the church, Cartesianism was warmly adopted by the Oratory, which saw in Descartes something of St Augustine, by Port Royal, which discovered a connexion between the new system and Jansenism, and by some amongst the Benedictines and the order of Ste Genevieve.Besides the logarithms and the calculating rods or bones, Napier's name is attached to certain rules and

**formulae**in spherical trigonometry.The rules are curious and interesting, but of very doubtful utility, as the

**formulae**are best remembered by the practical calculator in their unconnected form.But not only is the combining power or valency (atomicity) of the elements different, it is also observed that one element may combine with another in several proportions, or that its valency may vary; for example, phosphorus forms two chlorides represented by the

**formulae**PC1 3 and PC1 51 nitrogen the series of oxides represented by the**formulae**N 2 0, NO, (N203), N 2 O 4, N205, molybdenum forms the chlorides MoC1 2, MoC1 3, MoC1 4, MoC1 5, MoCls(?), and tungsten the chlorides WC1 2, WCl 4, WC1 5, WC16.Other features reminiscent of the original barbarous rites in the primitive caverns of the East, no doubt also occupied a place in the cult; bandaging of eyes, binding of hands with the intestines of a fowl, leaping over a ditch filled with water, witnessing a simulated murder, are mentioned by the Pseudo-Augustine; and the manipulation of lights in the crypt, the administration of oaths, and the repetition of the sacred

**formulae**, all contributed toward inducing a state of ecstatic exaltation.The agreement of the values of H with those of Griffiths and Dieterici at low temperatures, and of the values of p with those of Regnault over the whole range, are a confirmation of the accuracy of the foregoing theory, and show that the behaviour of a vapour like steam may be represented by a series of thermodynamically consistent

**formulae**, on the assumption that the limiting value of the specific heat is constant, and that the isothermals are generally similar in form to those of other gases and vapours at moderate pressures.