# Formula Sentence Examples

Every morning before I went to school I had chores to do, which began with mixing up the

**formula**and feeding the calves.The students could not figure out the math problem until their teacher provided them with a helpful

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

**formula**instead.The chemical

**formula**shows all the elements and compounds involved in the reaction.In point of fact, bourgeois democracy is the political

**formula**for free trade, nothing more.Aiden's mother sent a big tub of

**formula**to daycare so he would not get hungry.I bought some lamb milk

**formula**.Bredig point to the salt possessing the double

**formula**.Cane-sugar has the

**formula**C12H22011.This

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

**formula**of naphthalene, C 1 oH 8; by fusing three, the hydrocarbons anthracene and phenanthrene, C14H10; by fusing four, chrysene, C18H12, and possibly pyrene, C16H1n; by fusing five, picene, C22 H 14.This

**formula**is valid between speeds of 37 and 77 m.This implied that in the benzene complex there was at least one carbon atom linked to three others, thus rendering Kekule's

**formula**impossible and Ladenburg's and Claus' possible.Van 't Hoff's

**formula**is equivalent to taking the frequency of dissociation as proportional to the square of the concentration of the molecules, and the frequency of recombination as proportional to the cube of the concentration of the ions.The forces between the ions of a strongly dissociated solution will thus be considerable at a dilution which makes forces between undissociated molecules quite insensible, and at the concentrations necessary to test Ostwald's

**formula**an electrolyte will be far from dilute in the thermodynamic sense of the term, which implies no appreciable intermolecular or interionic forces.AdvertisementThe properties of caoutchouc clearly show, however, that its actual molecular structure is considerably more complex than is represented by the empirical

**formula**, and that it is to be regarded as the polymer of a terpene or similar hydrocarbon and composed of a cluster of at least ten or twenty molecules of the**formula**C5H8.The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete symmetric function of that weight; it is denoted by h.; it is connected with the elementary functions by the

**formula**1 7r1l7r2!7r3!The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic

**formula**we can at once write down an operation**formula**.It dissolves in an excess of alkali to form plumbites of the general

**formula**Pb(OM) 2.Steinmetz's

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

**formula**of this type has no physical significance, and cannot be accepted as an equation to the actual curve of W and B, it is, nevertheless, the case that by making the index e =1.6, and assigning a suitable value to r t, a**formula**may be obtained giving an approximation to the truth which is sufficiently close for the ordinary purposes of electrical engineers, especially when the limiting value of B is neither very great nor very small.Shimizu 3 indicate that Steinmetz's

**formula**holds for nickel and annealed cobalt up to B =3000, for cast cobalt and tungsten steel up to B =8000, and for Swedish iron up to B =18,000, the range being in all cases extended at the temperature of liquid air.He showed that there was, on the whole, a fair agreement between the values determined ballistic ally and those given by the

**formula**B = 871-F.If the pull is measured in pounds and the area in square inches, the

**formula**may be written B =1317 X iI P/S +H.The range of + B within which Steinmetz's

**formula**is applicable becomes notably increased at low temperature.AdvertisementSteinmetz's

**formula**applies only for very weak inductions when the alloys are at the ordinary temperature, but at the temperature of liquid air it becomes applicable through a wide range of inductions.Owing to the difficulty of determining the magnetization I and the susceptibility K with accuracy, it has not yet been possible to submit this

**formula**to a quantitative test, but it is said to afford an indication of the results given by actual experiment.It is pointed out that this

**formula**may be used as a temperature correction in magnetic determinations carried out in air.His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical

**formula**of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities.To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful

**formula**of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.A

**formula**was found by Bynkershoek in his De dominio maxis (1702) for the restriction of dominion over the sea to the actual distance to which cannon range could protect it.The inherent difficulties of Realism led to a J variety of attempts to reach a more satisfactory

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

**formula**.In the case of division we get two kinds of arithmetical

**formula**, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result.In the second place, particular cases lay the foundation for the general

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

**formula**.The binomial theorem gives a

**formula**for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.Assuming the above

**formula**to represent guncotton, there is sufficient oxygen for internal combustion without any carbon being left.He found out the

**formula**for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines.The Deep Sanderson

**formula**junior is now nearing completion and is looking exquisite.There may be an advantage to the divorce action being filed in a certain jurisdiction; for example a couple may not want to file in a community property state if they have agreed on a different

**formula**for dividing their marital property.Strecker to be trichloracetoacrylic acid, was more favourably explained by his

**formula**than by Ladenburg's.From these results Baeyer concluded that Claus'

**formula**with three para-linkings cannot possibly be correct, for the Q2.5 dihydroterephthalic acid undoubtedly has two ethylene linkages, since it readily takes up two or four atoms of bromine, and is oxidized in warm aqueous solution by alkaline potassium permanganate.The stronger argument against the ethylenoid linkages demanded by Kekule's

**formula**is provided by the remarkable stability towards oxidizing and reducing agents which characterizes all benzenoid compounds.In this

**formula**, the so-called " centric**formula**," the assumption made is that the fourth valencies are simply directed towards the centre of the ring; nothing further is said about the fourth valencies except that they exert a pressure towards the centre.Claus maintained that Baeyer's view was identical with his own, for as in Baeyer's

**formula**, the fourth valencies have a different function from the peripheral valencies, being united at the centre in a form of potential union.Marckwald (Ann., 93, 74, 331; 1894, 79, 14) has supported this

**formula**from considerations based on the syntheses of the quinoline ring.Bamberger (a strong supporter of the centric

**formula**), have shown that the nature of the substituent groups influences the distribution of the fourth valencies; therefore it may be con-?A similar contradiction apparently exists with regard to the specific volume, for while benzene has a specific volume correspinding to Claus'

**formula**, toluene, or methylbenzene, rather points to Kekule's.Stohmann of Leipzig; and the new data and the conclusions to be drawn from them formed the subject of much discussion, Briihl endeavouring to show how they supported Kekule's

**formula**, while Thomsen maintained that they demanded the benzene union to have a different heat of combustion from the acetylene union.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.Experiments showed that the second acid was much more difficult to esterify than the first, pointing to the conclusion that Claus'

**formula**for benzene was more probable than Kekule's.Ladenburg's prism

**formula**would give two enantiomorphic ortho-di-substitution derivatives; while forms in which the hydrogen atoms are placed at the corners of a regular octahedron would yield enantiomorphic tri-substitution derivatives.The octahedral

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

**formula**(2), being unsymmetrical, is impossible.It remains, therefore, to consider Erlenmeyer's

**formula**and those derived from the centric hypothesis.The centric

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

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

**formula**C14H10, are to be regarded as formed by the fusion of three.Thus a comparison of numbers occurring in a table of squares 1 2 =1 112=121 2 2= 4 122=144 3 2 =9 132=169 suggests the

**formula**(A+a)2=A2+2Aa+a2.Now suppose that the

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

**formula**(2) is also true for the nth power of A-I-a.Thus we get a

**formula**for the sum of n terms of a series such as 2.4.6+4.6.8+..., or 6.8.10.12+8.10.12.14+...The general

**formula**(which is established by more advanced methods) is 4.Whatever value is to be attached to Mendel's observation of the breaking up of self-fertilized hybrids of cultivated varieties into the two original parent forms according to the

**formula**" 'PP, 2PN, INN," it cannot be considered as more than a contribution to the extensive investigation of heredity which still remains to be carried out.From the general

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

**formula**expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory.Fresnel (1815), who gave a

**formula**equivalent to (5) below.If 8 and 4' denote the angles with the normal made by the incident and diffracted rays, the

**formula**(5) still holds, and, if the deviation be reckoned from the direction of the regularly reflected rays, it is expressed as before by (0+0), and is a minimum when 8 = 0, that is, when the diffracted rays return upon the course of the incident rays.This is the ordinary

**formula**for a reflecting plane grating, and it shows that the spectra are formed in the usual directions.The integration of the several terms may then be effected by the

**formula**e y dy =r(4+2)=(4 - i)(4-2)...Peligot's results, though called in question by Berzelius, have been amply confirmed by all subsequent investigators; only now, on theoretical grounds, first set forth by Mendeleeff, we double Peligot's atomic weight, so that U now signifies 240 parts of uranium, while UO 3 stands as the

**formula**of the yellow oxide, and UO 2 as that of Berzelius's metal.Stannic sulphide, SnS 2, is obtained by heating a mixture of tin (or, better, tin amalgam), sulphur and sal-ammoniac in proper proportions in the beautiful form of aurum musivum (mosaic gold) - a solid consisting of golden yellow, metallic lustrous scales, and used chiefly as a yellow "bronze" for plaster-of-Paris statuettes, &c. The yellow precipitate of stannic sulphide obtained by adding sulphuretted hydrogen to a stannic solution readily dissolves in solutions of the alkaline sulphides to form thiostannates of the

**formula**M 2 SnS 31 the free acid, H2SnS3, may be obtained as an almost black powder by drying the yellow precipitate formed when hydrochloric acid is added to a solution of a thiostannate.This colouring power is due to the presence of polychlorite, a substance whose chemical

**formula**appears to be C4 8 1-160018, and which may be obtained by treating saffron with ether, and afterwards exhausting with water.The smallest empirical

**formula**for cellulose (q.v.) may certainly be written C6H1005.If guncotton be correctly represented by the

**formula**C 6 H 7 0 2 (NO 3) 3, it should contain a little more than 14% of nitrogen.Busch (Ber., 1905, 38, pp. 856, 4049) has isolated a series of bridged ring compounds which he describes as endo-iminodihydrotriazoles, the triphenyl derivative (annexed

**formula**) being prepared by condensing triphenylaminoguanidine with formic acid.In reply to Rudolf Hospinian's Concordia discors (1607), he wrote a work, rich in historical material but one-sided in its apologetics, Concordia concors (1614), defending the

**formula**of Concord, which he regarded as inspired.Another work, based on the

**formula**of Concord, was entitled Loci communes theologici.His

**formula**was "to royalize France and to nationalize the monarchy."I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and

**formula**(II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.Its dental

**formula**(as that of Tragulus) is i., c. m.1 =34.The first term includes simple sugars containing two to nine atoms of carbon, which are known severally as bioses, trioses, tetroses, pentoses, hexoses, &c.; whilst those of the second group have the

**formula**C12H22011 and are characterized by yielding two monosaccharose molecules on hydrolysis.In addition trisaccharoses are known of the

**formula**C13H32016; these on hydrolysis yield one molecule of a monosaccharose and one of a disaccharose, or three of a monosaccharose.According to Mayer, the

**formula**of convolvulin is C34HSO016, and that of jalapin C31HS0016.To the bishop's

**formula**, "I separate thee from the church militant and the church triumphant," Savonarola replied in firm tones, "Not from the church triumphant; that is beyond thy power."C. von Hahn as the Aryan Expulsion and Return

**formula**, which counts among its representatives such heroes as Perseus, Cyrus, Romulus and Remus, Siegfried, and, as Alfred Nutt has pointed out, Arthur himself.Spencer tries to express in a sweeping general

**formula**the belief in progress which pervaded his age, and to erect it into the supreme law of the universe as a whole.The

**Formula**of Concord (1577),which gave to the whole Lutheran Church of Germany a common doctrinal system, declined to accept the Calvinistic position that man's condemnation as well as his salvation is an object of divine predestination.Methane and its homologues give origin to the " paraffin " or " fatty series " of the general

**formula**C,H 2, ,+ 1 000H, ethylene gives origin to the acrylic acid series, C n H 27, - 1 000H, and so on.Dibasic acids of the paraffin series of hydrocarbons have the general

**formula**C n H 2 (000H) 2 "; malonic and succinic acids are important members.By uttering a sacred

**formula**the good spirit throws the evil one into a state of confusion for a second 3000 years, while he produces the archangels and the material creation, including the sun, moon and stars.The summit C of the branch BCD occurs at a percentage exactly corresponding to the

**formula**Hg 2 T1.Potassium and sodium readily dissolve in ethyl alcohol with the production of alcoholates of the

**formula**C2 H5 OK(Na).In the extreme case when e=1, the prolate ellipsoid becomes a long thin rod, and then the capacity is given by C 1 = a/log e 2a/b (io), which is identical with the

**formula**(2) already obtained.This

**formula**is important in connexion with the capacity of electric cables, which consist of a cylindrical conductor (a wire) enclosed in a conducting sheath.Many complicated expressions have been suggested by subsequent writers in the attempt to represent the continuity of the gaseous and liquid states in a single

**formula**, but these are of a highly empirical nature, and beyond the scope of the present inquiry.This simple result is generally true, and the corresponding expressions for G" and J" are valid, provided that c - b in

**formula**(17) is a function of the temperature only.They are not conceived of in any anthropomorphic form, their sex even may often be indeterminate ("sive mas, sive femina" is the constantly recurring

**formula**of prayer), but the sphere of action of each is clearly marked and an appeal to a spirit outside his own special sphere would never even be thought of.In conjunc tion with Andrea and Selnecker he induced the Lutherans of Saxony and Swabia to adopt the

**Formula**Concordiae and so become one body.Schmalkalden, drawn up by Luther in 1536, Luther's catechisms, and the

**Formula**of Concord which was an attempt to settle doctrinal divisions promulgated in 1580, sum up what is called " the confessional theology of Lutheranism."It shows the influence of Arminian theology against Calvinism, which was vigorously upheld in the Quin-particular

**formula**, put forward by the synod of Dort in 1619 to uphold the five points of Calvinism, after heated discussion, in which English delegates took part, of the problems of divine omniscience and human free-will.True, the impression conveyed by the attitude of the Catholic party at the second Diet of Spires had served to awaken the feeling for solidarity among the Evangelicals there assembled; and on the 22nd of April they had even secured the basis for a provisional alliance in the shape of a

**formula**drawn up by Bucer and dealing with the Lord's Supper.The dental

**formula**is i.This

**formula**must have been known to Vieta in 1593.There are several corrections of the

**formula**O =W/(W - W1) necessary to the accurate expression of the density.The pipette having been carefully dried, the process is repeated with pure alcohol or with proof spirits, and the strength of any admixture of water and spirits is determined from the corresponding number of drops, but the

**formula**generally given is not based upon sound data.The "service of God and his majesty" was the

**formula**which expressed the belief of the sovereign and his subjects.He has been classed with Arians, but he endorses in his own way the homoousian

**formula**, and denounces Arius as " Christi gloriae incapacissimus."Such a simple

**formula**is only possible because the salts of sea-water are of such uniform composition throughout the whole ocean that the chlorine bears a constant ratio to the total salinity as newly defined whatever the degree of concentration.The temperature of maximum density of sea-water of any specific gravity was found by Knudsen to be given with sufficient accuracy for all practical purposes by the

**formula**0 = 3.950.2660 -0, where 0 is the temperature of maximum density in degrees centigrade.The number of molecules of the first kind of gas, whose components of velocity lie within the ranges between u and u+du, v and v+dv, w and w+dw, will, by

**formula**(5), be v?l (h 3 m 3 /7 3)e hm (u2+v2+w2)dudvdw (9) per unit volume.To examine what is meant by a zero value of n we refer to

**formula**(15).If a solid body is regarded as an aggregation of similar atoms each of mass m, its specific heat C is given, as in

**formula**(19) by C = i (n+3) R/Jm.In the

**formula**used by Synesius (4,0) which is to be found in Bingham's Antiquities, we already find the attention of magistrates specially called to the censured person.In conditional baptism the Latins, since about the year 1227, use the

**formula**, " If thou art not baptized, then do I baptize thee," &c. The Latins further insist on a strict observance of the traditional matter and form.Its compound with calcium chloride has the

**formula**CaC1 2.4CH 3.It passes over equal spaces in equal times, but falls with an accelerating velocity according to the

**formula**h = zgt 2, where h is the height fallen through, g the force of gravity, and t the time of flight.The

**formula**for length of scale is, length = sighting radius X tangent of the angle of elevation.None are to partake of it save those who have been "baptized in the name of the Lord" (an expression which is of interest in a document which prescribes the threefold

**formula**).On the Protestant side the identity is still clear in the Lutheran

**Formula**of Concord (1577).It has the exact vapour-density corresponding to the

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

**formula**.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.The deduction of the

**formula**tab, where a and b are numbers, should be regarded as a later step.This involves the use of Cartesian co-ordinates, and leads to important general formulae, such as Simpson's

**formula**.The numerical result obtained by applying a

**formula**to particular data will generally not be exact.The length of the arc of a circle, for instance, is known if the length of the chord and its distance from the middle point of the arc are known; but it may be more convenient in such a case to use a

**formula**such as Huygens's rule than to obtain a more accurate result by means of trigonometrical tables.Such a

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

**formula**is a deduction from a general**formula**, considered later (§ 58), and may be verified in various ways.Hence, by (i) and (ii), the

**formula**holds for this figure.By (i) and (iii), the

**formula**holds for each of these figures; and therefore it holds for the prismoid as a whole.The first of these is a particular case of the prismoidal

**formula**(§ 58).The area is then given by the

**formula**of § 25 (ii).This is Simpson's

**formula**.This is Simpson's second

**formula**.It may be deduced from the

**formula**given above.Simpson's (first)

**formula**, for instance, holds for f = I, and is obtained by taking p = i and ignoring differences after 52up.Thus we find from (i) that Simpson's second

**formula**, for the case where the top is a parabola (with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us).This might have been deduced directly from Simpson's first

**formula**, by a series of eliminations.The same

**formula**will also hold (§ 52) for any cubic trapezette through the points.It follows from §§ 48 and 51 that, if V is a solid figure extending from a plane K to a parallel plane L, and if the area of every cross-section parallel to these planes is a quadratic function of the distance of the section from a fixed plane parallel to them, Simpson's

**formula**may be applied to find the volume of the solid.This

**formula**applies to such figures as the cone, the sphere, the ellipsoid and the prismoid.The

**formula**is known as the prismoidal**formula**.If u is an algebraical function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a

**formula**Aovo-1--yivi+..This ordinate will be an algebraical function of x, and we can again apply a suitable

**formula**.The areas of the sides for which 0 and x=xo+2h, and of the section by the plane x=xo+h, may be found by Simpson's second

**formula**; call these Ao and A2, and Al.Hence Simpson's

**formula**applies, and the volume is lh(A0-I-4A1+ A2).The process is simplified by writing down the general

**formula**first and then substituting the values of u.The volume of a frustum of a cone, for instance, can be expressed in terms of certain magnitudes by a certain

**formula**; but not only will there be some error in the measurement of these magnitudes, but there is not any material figure which is an exact cone.The

**formula**may, however, be used if the deviation from conical form is relatively less than the errors of measurement.In the case of mathematical functions certain conditions of continuity are satisfied, and the extent to which the value given by any particular

**formula**differs from the true value may be estimated within certain limits; the main inaccuracy, in favourable cases, being due to the fact that the numerical data are not absolutely exact.Applying Simpson's

**formula**to each of these, we have A -__9= *h(uo + 4 14 1 + u2) + 3 h (14 2 + 4u3 + u4) + ..The legitimacy of this assumption, and of the further assumption which enables the area of the new figure to be expressed by an approximate

**formula**instead of by an exact**formula**, must be verified in every case by reference to the actual differences.The alternative method, therefore, consists in taking a simple

**formula**, such as the trapezoidal rule, and correcting it to suit the mutual relations of the differences.Their generalization is given by the Euler-Maclaurin

**formula**= I, = 0, = 0, = 0 .Either or both of the expressions K and L will have to be calculated by means of the

**formula**of § 84; if this is applied to both expressions, we have a**formula**which may be written in a more general form f f 4 u4(x, y) dx dy = u dx dy.The Euler-Maclaurin

**formula**(§ 75) assumes that the bounding values of u', u"',..To find the quantity of timber in a trunk with parallel ends, the areas of a few sections must be calculated as accurately as possible, and a

**formula**applied.The

**formula**applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56).He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p. His value of the velocity in air is therefore U = iJ (p ip.) (Newton's

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

**formula**].This

**formula**was very nearly confirmed for hydrogen, carbon dioxide and nitrous oxide.Its introduction into the baptismal

**formula**(in the Greek Church it is pronounced after the name of each person of the Trinity) is probably later.Hence the

**formula**is more useful in the form w = (w i +w2)1 2 / (Kd -1 2) = (wl +w 2)lr/ (K -lr) where k= (wl+w2-1-w3)lr/w3 is to be deduced from the data of some bridge previously designed with the same working stresses.The reciprocal figure for any loaded frame is a complete

**formula**for the stress on every member of a frame of that particular class with loads on given joints.About 15% of a volatile oil is obtained by distilling cubebs with water; after rectification with water, or on keeping, this deposits rhombic crystals of camphor of cubebs, C 15 H 26 O; cubebene, the liquid portion, has the

**formula**C15HV4.It is evident, therefore, that the request for a definition of Ultramontanism cannot be answered with a concise

**formula**, but that the varied character of its manifestations necessitates a more detailed examination of its peculiar objects.But about the r3th century the Roman

**formula**was altered, and the council of Trent (1551) declared that the "form" and power of the sacrament of penance lay in the words Ego to absolvo, &c., and that the accompanying prayers are not essential to it.On hearing from Rome, Cyril at once held a synod and drew up a doctrinal

**formula**for Nestorius to sign, and also twelve anathemas covering the various points of the Nestorian dogmatic. Nestorius, instead of yielding to the combined pressure of his two great rivals, merely replied by a counter excommunication.The commotion which had been thus raised did not so easily subside in the more eastern section of the church; the Antiochenes continued to maintain for a considerable time an attitude of antagonism towards Cyril and his creed, and were not pacified until an understanding was reached in 433 on the basis of a new

**formula**involving some material conce9sions by him.His seeming inconsistencies are reconciled to apprehension, not by a

**formula**of the intellect, but by the many-sidedness of a highly impressible nature.MacCullagh's hands the correct equations were derived from a single energy

**formula**by the principle of least action; and while the validity of this dynamical method was maintained, it was frankly admitted that no mechanical analogy was forthcoming.The polarization itself is determined from the electric force (P,Q,R) by the usual statical

**formula**of linear type which becomes tor an isotropic medium (.f',g',h') = c2(P,Q,R), because any change of the dielectric constant K arising from the convection of the material through the aether must be independent of the sign of v and therefore be of the second order.In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical

**formula**composed of a single power of v, say v m, the integration can be effected which replaces the summation in (to), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f (v)or v m lk or its equivalent Cr, where r is the retardation.Integrating (27) again, (31) y =g(zTt2t 2) = zgt(T -t); and denoting T-t by t', and taking g= 32f/s2,) y =16tt', (32 which is Colonel Sladen's

**formula**, employed in plotting ordinates.The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical

**formula**, as explained in the article Armour Plates.But no

**formula**has yet been invented, derived on theoretical principles from the physical data, which will assign by calculation a definite magnitude to 3.The

**formula**shows that except for numbers of the form (3n 2 n) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity.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.The olefines may be synthetically prepared by eliminating water from the alcohols of the general

**formula**CnH2n+1 OH, using sulphuric acid or zinc chloride generally as the dehydrating agent, although phosphorus pentoxide, syrupy phosphoric acid and anhydrous oxalic acid may frequently be substituted.This fact, coupled with the determination of the vapour density of the gas, establishes the molecular

**formula**CO.The current theological

**formula**for this two-sided position is that the prophets are at once preachers of the law and forerunners of the gospel; and, as it is generally assumed that they found the law already written, their originality and real importance is made to lie wholly in their evangelical function.It will be evident even from this rapid sketch, necessarily confined to a few of the most cardinal points, that Hebrew prophecy is not a thing that can be defined and reduced to a

**formula**, but was a living institution which can only be understood by studying its growth and observing its connexion with the historical movements with which its various manifestations were bound up. Throughout the great age of prophecy the most obvious formal character that distinguished it was that the 1 One might say from the days of Habakkuk.This equation defines log x for positive values of x; if o the

**formula**ceases to have any meaning.It will be observed that in the first process the value of the modulus is in fact calculated from the

**formula**.The logarithm is then obtained by use of the

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

**formula**arrived at by Cauchy was n= A-FB+?If suitable values are chosen for these constants, the

**formula**can be made to represent the dispersion of ordinary transparent media within the visible spectrum very well, but when extended to the infra-red region it often departs considerably from the truth, and it fails altogether in cases of anomalous dispersion.There are also grave theoretical objections to Cauchy's

**formula**.Ketteler and known as the Ketteler-Helmholtz

**formula**, has been much used in calculating dispersion, and expresses the facts with remarkable accuracy.P. Drude has obtained a similar

**formula**based on the electromagnetic theory, thus placing the theory of dispersion on a much more satisfactory basis.As A m is a wave-length corresponding to an absorption band, this

**formula**can be used to find values of A m which satisfy the observed values of n within the region of transparency, and so to determine where the absorption bands are situated.His refusal to subscribe unconditionally to the rigid

**formula**of belief adopted by the theologians of Tubingen permanently closed against him the gates of his alma mater.Rohner (Ber., 1909, 42, p. 4 0 93) obtained a yellowish white oxide, of the

**formula**I 4 0 9, which they regard as an iodate of tervalent iodine, Millon's oxide being considered a basic iodate.On the next topic of importance, the primacy of the pope, the project of union nearly suffered shipwreck; but here a vague

**formula**was finally constructed which, while acknowledging the pope's right to govern the church, attempted to safeguard as well the rights of the patriarchs.And although the God of light himself now took to the field, and with the help of new aeons (the spirit of life, &c.) inflicted total defeat upon Satan, and set the ' A is spoken of in the

**formula**of abjuration, and an Epistola ad virginem Menoch by Augustine.The milk-dentition, and even the early condition of the permanent dentition, is formed on the same general type as that of Sus, except that certain teeth are absent, the

**formula**being 13 i cl, total 34; but as age advances all the teeth have a tendency to disappear, except the canines and the posterior molars, but these, which in some cases are the only teeth left in the jaws, attain an extraordinary development.Again, like the hierophant, the sycophant publicly pronounced the

**formula**of exclusion of certain unworthy persons from the celebration of the mysteries of the fig.A synopsis of the leading

**formula**connected with the circle will now be given.Archimedes's process of unending cycles of arithmetical operations could at best have been expressed in his time by a " rule" in words; in the 16th century it could be condensed into a "

**formula**."Accordingly, we find in Vieta a

**formula**for the ratio of diameter to circumference, viz.This Rule Is Conveniently Expressed By The

**Formula**(X 9) R, In Which X Denotes The Date, And The Symbol R Denotes That The Remainder, Which Arises From The Division Of X 9 By 28, Is The Number Required.The Symbol X, Employed In The

**Formula**At The Top Of The Column, Denotes The Number Of Centuries, That Is, The Figures Remaining After The Last Two Have Been Struck Off.This number is called the Golden Number, either because will therefore be given by the

**formula**(x 1 53) r, that is to say, add 3 totlzedate,divide the sum by 15, and the remainder is the year of the indiction.In This Case, As The

**Formula**Supposes The Intercalation Already Made, The Resulting Letter Is That Which Applies After The 29Th Of February.In Order To Investigate A

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

**Formula**N (N 1).The Value Of L Is Always Given By The

**Formula**For The Dominical Letter, And P And 1 Are Easily Deduced From The Epact, As Will Appear From The Following Considerations.P= 54 E 1=57 E, Or (57 E) 7 R' By Substituting One Or Other Of These Values Of P And 1, According A; The Case May Be, In The

**Formula**P = P (L L), We Shall Have P, Orthe Number Of Days From The 21St Of March To Easter Sunday.This

**Formula**Gives The Following Rule For Calculating The Date Of The Commencement Of Any Year Of The Hegira, According To The Gregorian Or New Style.Appended to the London edition of the solar and lunar tables are two short tracts - the one on determining longitude by lunar distances, together with a description of the repeating circle (invented by Mayer in 1752), the other on a

**formula**for atmospheric refraction, which applies a remarkably accurate correction for temperature.He describes himself when he says, "The student of Christian doctrine, because he strives after exactness of phrase, because he is conscious of the inadequacy of any one human

**formula**to exhaust the truth, will be filled with sympathy for every genuine endeavour towards the embodiment of right opinion.The dental

**formula**varies from 1, c. $, 3 (total 28) in the hares and rabbits to i.Possibly the European forms, in which the dental

**formula**has been given as i., c. S, p, m.Fibroin, which is analogous to horn, hair and like dermal products, constitutes about 75 to 82% of the entire mass, and has a composition represented by the

**formula**C15H23 506.Its

**formula**is C15H25N508.If within the range5100-3700A, the constants are determined once for all, the

**formula**seems capable of giving by interpolation results accurate to o 2 A, but as a rule the range to which the**formula**is applied will be much less with a corresponding gain in the accuracy of the results.Balmer's

**formula**received a striking confirmation when it was found to include the ultra-violet lines which were discovered by Sir William Huggins' in the photographic spectra of stars.The most complete hydrogen spectrum is that measured by Evershed 8 in the flash spectrum observed during a total solar eclipse, and contains thirty-one lines, all of which agree with considerable accuracy with the

**formula**, if the frequency number n is calculated correctly by reducing the wave-length to vacuo.9 It is a characteristic of Balmer's**formula**that the frequency approaches a definite limit as s is increased, and it was soon discovered that in several other spectra besides hydrogen, series of lines could be found, which gradually come nearer and nearer to each other as they become fainter, and approach a definite limit.Such series ought all to be capable of being represented by a

**formula**resembling that of Balmer, but so far the exact form of the series has not been established with certainty.C. Pickering' used in a special case, presently to be referred to, was put into a more general form by Thiele, who, however, assumes N to have the same value for all spectra, and not obtaining sufficient agreement, rejects the

**formula**.If we compare Balmer's

**formula**with the general equation of Ritz, we find that the two can be made to agree if the ordinary hydrogen spectrum is that of the side branch series and the constants a', b, c and d are all put equal to zero.'A discussion of band spectra on a very broad basis was given by Thiele,' who recommends a

**formula**- q +qi(s+c)+ +qr(s+c)r n in the discharge, except within the region of the kathode glow.Three acids of this empirical

**formula**are known, viz.The widow then loosens and removes the shoe, throwing it some distance, and spits on the ground, repeating thrice the Biblical

**formula**"So shall it be done," &c. Ilalisah, which is still common among orthodox Jews, must not take place on the Sabbath, a holiday, or the eve of either, or in the evening.This is used as the sulphate, which has the empirical

**formula**of (C15H21N302)2, H 2 SO 4, plus an unknown number of molecules of water.Later, the Variata of 1540 became the creed of the lVlelanchthonians and even of the Crypto-calvinists; so the framers of the

**Formula**of Concord, promulgated in 1580, returned to the text handed in at the Diet.The textus receptus is that of the

**Formula**of Concord, the divergent Latin and German forms being equally binding.Its antiquity is attested by the symbol and

**formula**used in its procedure, the lance (hasta) as the sign of true ownership, the oath or wager (sacramentum), the ancient**formula**for recovery of property or assertion of liberty.That orthoboric acid is a tribasic acid is shown by the formation of ethyl orthoborate on esterification, the vapour density of which corresponds to the molecular

**formula**B(0C2H5)3; the molecular**formula**of the acid must consequently be B(OH) 3 or H 3 B0 3.But the free-traders did not like Mr Balfour's

**formula**as to reversing the traditional fiscal policy of import taxes for revenue only.As regards the recent forms, the dentition in the fully adult animal consists only of incisors and cheek-teeth, the

**formula**being i.Since this party in the course of years came more and more into sympathy with the representatives of the Nicene party, the Homoousians, and notably with Athanasius, the much-disputed

**formula**became more and more popular, till the council summoned in 38L at Constantinople, under the auspices of Theodosius the Great, recognized the Nicene doctrine as the only orthodox one.The seal is that of the fisherman's ring, hence the

**formula**of conclusion, "Datum Romae, sub annulo Piscatoris."It often happens that analyses of brown iron ores reveal a larger proportion of water than required by the typical

**formula**of limonite, and hence new species have been recognized.Schmidt xanthosiderite, from 1'avOos (yellow) and otSnpos (iron), contains Fe 2 0(OH) 4, or Fe 2 0 3.2H 2 O; whilst the bog ore known as limnite, from At vn (marsh) has the

**formula**Fe(OH) 3, or Fe203.3H20.Its vapour density at temperatures above 750 corresponds to the

**formula**AlCl 3 j below this point the molecules are associated.According to the

**formula**FeS2, pyrites contains theoretically 46.67% of iron and 53.33 of sulphur.For instance, following Krupp's

**formula**, the side and barbette armour of war-vessels is now generally if not universally made of nickel steel containing about 3.25% of nickel, 0.40% of carbon, and 1.50% of chromium, deeply carburized on its impact face.The monovalent iodine, for instance, is transformed by heating into an allotropic form, corresponding to the

**formula**I, whereas ordinary iodine answers to I 2.Now, in this case, the first definition expresses much better the whole chemical behaviour of ozone, which is that of "energetic" oxygen, while the second only includes the fact of higher vapour-density; but in applying the first definition to organic compounds and calling isobutylene "butylene with somewhat more energy" hardly anything is indicated, and all the advantages of the atomic conception - the possibility of exactly predicting how many isomers a given

**formula**includes and how you may get them - are lost.It does not, however, satisfy the Socialists, whose

**formula**is one man, one vote.This

**formula**includes the vast majority of cardiac cases.When the

**formula**cited is not used, other somewhat nebulous expressions are sometimes employed, as, for example, that.If Y and Z are groups which behave very differently, then there is apparently no tautomerism and a definite

**formula**can be given to the compound.The conductivity of the same bars was independently determined by the method of Forbes, employing an ingenious

**formula**for the heat-loss in place of Newton's law.He was one of the most influential supporters of the

**Formula**Consensus Helvetica, drawn up chiefly by Johann Heinrich Heidegger (1633-1698), in 1675, and of the particular type of Calvinistic theology which that symbol embodied, and an opponent of the theology of Moses Amyraut and the school of Saumur.During the next forty years of his life he enjoyed great influence in Geneva as the advocate of a more liberal theology than had prevailed under the preceding generation, and it was largely through his instrumentality that the rule obliging ministers to subscribe to the

**Formula**Consensus Helvetica was abolished in 1706, and the Consensus itself renounced in 1725.Its vapour density has been determined by Nilson and Pettersson, and corresponds to the molecular

**formula**BeC12.The total number of teeth is 34, with the

**formula**i.3.Two acids are known corresponding to this

**formula**, normal butyric acid, CH 3 CH 2 CH 2.Lovering's general

**formula**suggests that the 4-month term is really less important than the 3-month term, but he gives no data for the latter at individual stations.They had been converted to Mahommedanism in the early times of the Arab conquest, but their knowledge of Islam did not go much beyond the

**formula**of the creed - "there is no god but God, and Mahomet is the apostle of God," - and they were ignorant of the law.No verbal

**formula**can really enclose the life of a people or an age, but we can best understand the significant ^ of the old Greek cities and the life they developed, when, looking at the history of mankind as a whole, we see the part played by reason, active and critical, in breaking down the barriers by which custom hinders movement, in guiding movement to definite ends, in dissipating groundless beliefs and leading onwards to fresh scientific conquests - when we see this and then take note that among the ancient Greeks such an activity of reason began in an entirely novel degree and that its activity in Europe ever since is due to their impulsion.From 1018 the relations of C, D, E become too complicated to be expressed by any

**formula**; sometimes all three agree together, sometimes all three are independent; in other places each pair in turn agree against the third.The occasional visitor to the tomb is reminded by its inscriptions of the many virtues of the dead man while he yet lived, and is charged, if he be come with empty hands, at least to pronounce the funerary

**formula**; it will indeed cost him nothing but the breath of his mouth !Often the

**formula**was written on a strip of rag or a scrap of papyrus and tied round the neck of the person for whom it was intended.Up to that time all the leaders had been united in accepting the naturalistic

**formula**, which was combined with an individualist and a radical tendency.The

**formula**employed by P is "and Pharaoh's heart was strong (pin)," or, "and Pharaoh made strong (p'in) his heart," as in E, but it is distinguished from E's phrase by the addition of "and he hearkened not unto them as Yahweh had spoken."The emperors Zeno and Anastasius had been strongly suspected of it, and the Roman bishops had refused to communicate with the patriarchs of Constantinople since 484, when they had condemned Acacius for accepting the

**formula**of conciliation issued by Zeno.One of Justinian's first public acts was to put an end to this schism by inducing Justin to make the then patriarch renounce this

**formula**and declare his full adhesion to the creed of Chalcedon.Titanotherium, of the Oligocene of the Dakotas and neighbouring districts, was a huge beast, with the hinder upper premolars similar in character to the molars, a pair of horn-cores, arising from the maxilla, overhanging the nose-cavity, four front and three hind toes, only twenty dorso-lumbar vertebrae, and an almost continuous and unbroken series of teeth, in which the canines are short; the dental

**formula**being i.Whether or no the so-called "fourth

**formula**" (Hahn, § 156) is to be ascribed to a continuation of this synod or to a subsequent but distinct assembly of the same year, its aim is like that of the first three; while repudiating certain Arian formulas it avoids the Athanasian shibboleth "homoousios."The chlorine is not completely precipitated by silver nitrate in nitric acid solution, the ionization apparently not proceeding to all the chlorine atoms. Thallic iodide, T11 3, is interesting on account of its isomorphism with rubidium and caesium tri-iodides, a resemblance which suggests the

**formula**T11 (12) for the salt, in which the metal is obviously monovalent.He was, as he called himself, a " mystic "; and his creed was too vague to be put into any

**formula**beyond a condemnation of atheism.Heidegger was the principal author of the

**Formula**Consensus Helvetica in 1675,which was designed to unite the Swiss Reformed churches, but had an opposite effect.A wide knowledge of the Old Testament supplies him with a text to illustrate one incident after another; and so deeply is he impressed with the correspondence between the life of Christ and the words of ancient prophecy, that he does not hesitate to introduce his quotations by the

**formula**" that it might be fulfilled which was spoken by the prophet."Metternich was anxious to secure an apparent unanimity of the powers to back the Austrian intervention in Naples, and every device was used to entrap the English representative into subscribing a

**formula**which would have seemed to commit Great Britain to the principles of the other allies.This derivation of the word Mass, which would connect it with the special

**formula**of dismissal still preserved in the Roman liturgy - Ite, missa est- once generally accepted, is now disputed.A graver mistake, according to some critics, is that Hegel, far from giving a law of progress, seems to suggest that the history of the world is nearing an end, and has merely reduced the past to a logical

**formula**.Its composition approaches the

**formula**CaOC1 2, and it is regarded as a double salt of calcium chloride and hypochlorite, which by the action of water splits up into a mixture of these salts.Their

**formula**in Na 2 CO 3, 10H 2 O, corresponding to 37% of dry sodium carbonate.Jefferson seems to have been the first discoverer of an exact

**formula**for the construction of mould-boards of least resistance for ploughs.The magnifying power is computed by the same

**formula**as in the case of the Gregorian telescope.This flamen also sacrificed on the 1st of May to Maia, who in an old prayer

**formula**(Gellius 13.23) was coupled with Volcanus as Maia Volcani.This quantity is now, as the result of further experiments, added to the values of h, and also represented in the

**formula**for the specific heat itself by the cubic term.It is a black powder, the composition of which is never quite definite, but approximates to the

**formula**given above.It is an advance on this when Heraclitus 2 opposes to the eyes and ears which are bad witnesses " for such as understand not their language " a common something which we would do well to follow; or again when in the incommensurability of the diagonal and side of a square the Pythagoreans stumbled upon what was clearly neither thing nor image of sense, but yet was endowed with meaning, and henceforth were increasingly at home with symbol and

**formula**.The result is a selflimiting dialectic. This higher dialectic is a logic. It is no accident that the first of the philosophical sophists, Gorgias, on the one hand, is Eleatic in his affinities, and on the other raises in the characteristic

**formula**of his intellectual nihilism' issues which are as much logical and epistemological as ontological.The one in the many, the

**formula**which lies at the base of the possibility of predication, is involved in the Socratic doctrine of general concepts or ideas.Yet from the points of view alike of an absolute pluralism, of a flux, and of a

**formula**of bare identity - and a fortiori with any blending of these principles sufficiently within the bounds of plausibility to find an exponent - all knowledge, because all predication of unity, in difference, must be held to be impossible.The

**formula**for an indefinite number of particular things in particular places at particular times, and all of them presentable in sensuous imagery of a given time and place, is not itself presentable in sensuous imagery side by side with the individual members of the group it orders.Plato replies with the doctrine of the interpenetration of ideas, obviously not of all with all, but of some with some, the

**formula**of identity in difference within thought itself.Once embarked upon such an analysis, where valid process from assured principles gave truth, Aristotle could find little difficulty in determining the

**formula**of demonstrative knowledge or science.The point of view is to be modified profoundly by what follows - by the doctrine of the class-concept behind the class, of the form or idea as the constitutive

**formula**of a substance, or, again, by the requirement that an essential attribute must be grounded in the nature or essence of the substance of which it is predicated, and that such attributes alone are admissible predicates from the point of view of the strict ideal of science.The question is as to membership of a class, and the dominant

**formula**concet.Syllogism as

**formula**for the exhibition of truth attained, and construction or what not as the instrumental process by which we reach the truth, have with writers since Hegel and Herbart tended to fall apart.The individualism with which he starts, howsoever afterwards mitigated by his doctrine of To Ti v eivac or eiSos constituting the individual in a system of intelligible relations, confined him in an inadmissible way to the subject-attribute

**formula**.The Aristotelian theory of the universal of science as secure from dependence on its instances and the theory of linking in syllogism remain a heritage for all later logic, whether accepted in precisely Aristotle's

**formula**or no.The method now proceeds on the basis of the first table to set f p rth the possible suggestions as to a general explanatory

**formula**for the quality in question.If one suggestion, and one alone, survives the process of attempted rejection it is the explanatory

**formula**required.It is the recipe for, and at the same time is, heat, much as H 2 O is the

**formula**for and is water.Had Bacon analysed bodies into their elements, instead of their qualities and ways of behaviour, he would have been the logician of the chemical

**formula**.Hobbes developed the nominalism which had been the hallmark of revolt against scholastic orthodoxy, and, when he brings this into relation with the analysis and synthesis of scientific A notable

**formula**of Bacon's Novum Organum ii.Hume derived from him the explanatory

**formula**of the association of ideas, 8 which is, however, still with Hobbes a fact to be accounted for, not a theory to account for facts, being grounded physically in " coherence of the matter moved."Berkeley, though at length the notions of spirits, acts and relations 6 give him pause, prefers the

**formula**which Hume expresses in the phrase that " some ideas are particular in their nature but general in their representation," 7 and the afterhistory of " abstraction " is a discussion of the conditions under which one idea " stands for " a group. Not from those for whom general ideas mean schematic concepts, not imageable.Apart from the adoption by Hume of the association of ideas as the explanatory

**formula**of the school - it had been allowed by Malebranche within the framework of his mysticism and employed by Berkeley in his theory of vision - there are few fresh notes struck in the logic of sensationalism.They had undermined the foundations of scientific certainty, and so far as the fecundity of contemporary science did not give them pause, were ready, notwithstanding the difference of their starting-point, to acquiesce in the

**formula**as well as the temper of Pyrrhonism.So again in determining the " import " of propositions, it is no accident that in all save existential propositions it is to the familiar rubrics of associationism - co-existence, sequence, causation and resemblance - that he refers for classification, while his general

**formula**as to the conjunctions of connotations is associationist through and through.It is a shorthand

**formula**of registration.But that truth is not what Mill expounds, nor is it capable of development within the limits imposed by the associationist

**formula**.Again the

**formula**of the Joint-Method, which contemplates the enumeration of cases " which have nothing in common but the absence of one circumstance," is ridiculously unsound as it stands.The latter becomes a mere memorandum or

**formula**of registration.It is the oscillation which Mill manifests between the conception of his

**formula**as it is actually applicable to concrete problems in practice, and the conception of it as an expression of a theoretical limit to practical procedure.Finally, it is on the whole in keeping with Mill's presuppositions to admit even in the case of the method of difference that in practice it is approximative and instructive, while the theoretical

**formula**, to which it aims at approaching asymptotically as limit, if exact, is in some sense sterile.Mill may well have himself conceived his methods as practically fruitful and normally convincing with the limiting

**formula**in each case more cogent in form but therewith merely the skeleton of the process that but now pulsed with life.The

**formula**of identity passed in another form to Herbart and therefore to Lotze.His general theory of knowledge deriving from Kant and Reid, and including among other things a contaminatio of their theories of perception, 3 in no way sustains or mitigates his narrow view of logic. He makes no effective use of his general

**formula**that to think is to condition.Schleiermacher's

**formula**obviously ascribes a function in knowledge to thought as such, and describes in a suggestive manner a duality of the intellectual and organic functions, resting on a parallelism of thought and being whose collapse into identity it is beyond human capacity to grasp. It is rather, however, a statement of a way in which the relations of the terms of the problem may be conceived than a system of necessity.There was scope for diversity of view and there was diversity of view, according as the vital issue of the

**formula**was held to lie in the relation of intellectual function to organic function or in the not quite equivalent relation of thinking to being.Finally it must not be forgotten that the host of writers who were in reaction against Hegelianism tended to take refuge in some

**formula**of correlation, as a half-way ho-use between that and formalism or psychologism or both, without reference to, and often perhaps without cdnsciousness of, the way in which historically it had taken shape to meet the problem held to have been left unresolved by Kant.Fichte cannot be said to have developed a logic, but this rhythm of thesis, antithesis and synthesis, foreshadowed in part for Fichte in Spinoza's

**formula**, " omnis determinatio est negatio," and significantly in Kant's triadic grouping of his categories, gave a cue to the thought of Hegel.To the formalist proper it was self-condemned in its pretension to develop the content of thought and its rejection of the

**formula**of bare-identity.The comparison of Sigwart with Lotze is instructive, in regard both to their agreement and their divergence as showing the range of the epistemological

**formula**.But every quaternion

**formula**is a proposition in spherical (sometimes degrading to plane) trigonometry, and has the full advantage of the symmetry of the method.The most important apparent exceptions to Raoult's law in dilute solutions are the cases, (I) in which the molecules of the dissolved substance in solution are associated to form compound molecules, or dissociated to form other combinations with the solvent, in such a way that the actual number of molecules n in the solution differs from that calculated from the molecular weight corresponding to the accepted

**formula**of the dissolved substance; (2) the case in which the molecules of the vapour of the solvent are associated in pairs or otherwise so that the molecular weight m of the vapour is not that corresponding to its accepted**formula**.The highest pressures recorded for cane-sugar are nearly three times as great as those given by van't Hoff's

**formula**for the gas-pressure, but agree very well with the vapour-pressure theory, as modified by Callendar, provided that we substitute for V in Arrhenius's**formula**the actual specific volume of the solvent in the solution, and if we also assume that each molecule of sugar in solution combines with 5 molecules of water, as required by the observations on the depression of the freezing-point and the rise of the boiling-point.Zeuner, at a later date (La Chaleur, p.441) employing the empirical,

**formula**pv= BO +Cp 25 for saturated steam, found the value S = 568, which further increased the discrepancy.Perry (Steam Engine, p. 580), assuming a characteristic equation similar to Zeuner's (which makes v a linear function of the temperature at constant pressure, and S independent of the pressure), calculates S as a function of the temperature to satisfy Regnault's

**formula**(10) for the total heat.If the steam at A were dry and saturated, we should have, assuming Regnault's

**formula**(to), H A -H D = 305 (0'-O), whence, if S = .475, we have zL = .3 0 5 (0 '- 0)-.It is evident that this is a very delicate method of determining the wetness z, but, since with dry saturated steam at low pressures this

**formula**always gives negative values of the wetness, it is clear that Regnault's numerical coefficients must be wrong.If we assume Regnault's

**formula**(10) for the total heat, we have evidently the simple relation S=0.305(0'-0)/(0"-o), supposing the initial steam to be dry, or at least of the same quality as that employed by Regnault.The method of deducing the specific heat from Regnault's

**formula**for the variation of the total heat is evidently liable in a greater degree to the objections which have been urged against his method of determining the specific heat, since it makes the value of the specific heat depend on small differences of total heat observed under conditions of greater difficulty at various pressures.The rate of increase of the total heat, instead of being constant for saturated steam as in Regnault's

**formula**, is given by the equation dH/d0 =S(1 - Qdp/d0).The

**formula**of Dalton would make the pressure increase in geometrical progression for equal increments of temperature.Observing that this rate of diminution is approximately as the square of the reciprocal of the absolute temperature, we see that the almost equally simple

**formula**log p=A+B/0 represents a much closer approximation to experiment.The values so found are necessarily erroneous if

**formula**(io) for the total heat is wrong.This leads immediately to the simple

**formula**loge (p/po) = (t/90t/6)L/R,.A

**formula**of this type has been widely employed by van't Hoff and others to calculate heats of reaction and solution from observations of solubility and vice versa.The

**formula**evidently applies to the vapour-pressure of the pure solvent as a special case, but Kirchhoff himself does not appear to have made this particular application of the**formula**.He proceeds to calculate from this expression the difference of vapour-pressures of ice and water in the immediate neighbourhood of the melting-point, but does not observe that the vapour-pressures themselves may be more accurately calculated for a considerable interval of temperature by means of

**formula**(23), by substituting the appropriate values of the latent heats and specific heats.They give the following numerical values The error of the

**formula**for water is less than t mm.