III. "Memoir on the Symmetric Functions of the Roots of certain Systems of two Equations." By ARTHUR CAYLEY, Esq., F.R.S. Received December 18, 1856.

(Abstract.)

The author defines the term roots as applied to a system of n-1 equations =0,=0, &c., where p, 4, &c., are quantics (i. e. rational and integral homogeneous functions) of the n variables (x, y, z, . .) and the terms symmetric functions and fundamental symmetric functions of the roots of such a system; and he explains the process given in Professor Schläfle's memoir, "Ueber die Resultante eines Systemes mehrerer algebraischer Gleichungen," Vienna Transactions, t. iv. (1852), whereby the determination of the symmetric functions of any system of (n-1) equations, and of the resultant of any system of n equations is made to depend upon the very simple question of the determination of the resultant of a system of n equations, all of them, except one, being linear. The object of the memoir is then stated to be the application of the process to two particular cases, viz. to obtaining the expressions for the simplest symmetric functions, after the fundamental ones of the following systems of two ternary equations, viz. first, a linear equation and a quadratic equation; and secondly, a linear equation and a cubic equation; and the author accordingly obtains expressions, as regards the first system, for the fundamental symmetric functions or symmetric functions of the first degree in respect to each set of roots, and for the symmetric functions of the second and third degrees respectively, and as regards the second system, for the fundamental symmetric functions or symmetric functions of the first degree, and for the symmetric functions of the second degree in respect to each set of roots.

IV. "Memoir on the Resultant of a System of two Equations." By ARTHUR CAYLEY, Esq., F.R.S. Received December 18,

is, it is well known, a function homogeneous in regard to the coefficients of each equation separately, viz. of the degree n in regard to the coefficients (a, b, . .) of the first equation, and of the degree m in regard to the coefficients (p, q,..) of the second equation; and it is natural to develope the resultant in the form kAP+k'A'P'+ &c., where A, A', &c. are the combinations (powers and products) of the degree n in the coefficients (a, b, . .), P, P', &c. are the combinations of the degree m in the coefficients (p, q, . .), and k, k', &c. are mere numerical coefficients. The object of the present memoir is to show how this may be conveniently effected, either by the method of symmetric functions, or from the known expression of the resultant in the form of a determinant, and to exhibit the developed expressions for the resultant of two equations, the degrees of which do not exceed 4. With respect to the first method, the formula in its best form, or nearly so, is given in the Algebra' of Meyer Hirsch, and the application of it is very easy when the necessary tables are calculated: as to this, see my "Memoir on the Symmetric Functions of the Roots of an Equation." But when the expression for the resultant of two equations is to be calculated without the assistance of such tables, it is, I think, by far the most simple process to develope the determinant according to the second of the two methods.

V. "Memoir on the Symmetric Functions of the Roots of an Equation." By ARTHUR CAYLEY, Esq., F.R.S. Received December 18, 1856.

(Abstract.)

There are contained in a work, which is not, I think, so generally known as it deserves to be, the Algebra' of Meyer Hirsch, some very useful tables of the symmetric functions up to the tenth degree of the roots of an equation of any order. It seems desirable to join to these a set of tables, giving reciprocally the expressions of the powers and products of the coefficients in terms of the symmetric functions of the roots. The present memoir contains the two sets of tables, viz. the new tables distinguished by the letter (a), and the tables of Meyer Hirsch distinguished by the letter (6); the memoir contains 2 c

VOL. VIII.

also some remarks as to the mode of calculation of the new tables, and also as to a peculiar symmetry of the numbers in the tables of each set, a symmetry which, so far as I am aware, has not hitherto been observed, and the existence of which appears to constitute an important theorem in the subject. The theorem in question might, I think, be deduced from a very elegant formula of M. Borchardt (referred to in the sequel), which gives the generating function of any symmetric function of the roots, and contains potentially a method for the calculation of the tables (6), but which, from the example I have given, would not appear to be a very convenient one for actual calculation.

VI. "Memoir on the Conditions for the Existence of given Systems of Equalities among the Roots of an Equation." By ARTHUR CAYLEY, Esq., F.R.S. Received December 18, 1856.

(Abstract.)

It is well known that there is a symmetric function of the roots of an equation, viz. the product of the squares of the differences of the roots, which vanishes when any two roots are put equal to each other, and that consequently such function expressed in terms of the coefficients and equated to zero, gives the condition for the existence of a pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of his earlier papers in the Philosophical Magazine,' that the like method could be applied to finding the conditions for the existence of other systems of equalities among the roots, viz. that it was possible to form symmetric functions, each of them a sum of terms containing the product of a certain number of the differences of the roots, and such that the entire function might vanish for the particular system of equalities in question; and that such functions expressed in terms of the coefficients and equated to zero would give the required conditions. The object of the present memoir is to extend this theory, and render it exhaustive by showing how to form a series of types of all the different functions which vanish for one or more systems of equalities among the roots; and in particular to obtain by the method distinctive conditions for all the different

systems of equalities between the roots of a quartic or a quintic equation, viz. for each system conditions which are satisfied for the particular system, and are not satisfied for any other systems, except, of course, the more special systems included in the particular system. The question of finding the conditions for any particular system of equalities is essentially an indeterminate one, for given any set of functions which vanish, a function syzygetically connected with these will also vanish; the discussion of the nature of the syzygetic relations between the different functions which vanish for any particular system of equalities, and of the order of the system composed of the several conditions for the particular system of equalities, does not enter into the plan of the present memoir. I have referred here to the indeterminateness of the question for the sake of the remark that I have availed myself thereof, to express by means of invariants or covariants the different systems of conditions obtained in the sequel of the memoir; the expressions of the different invariants and covariants referred to are given in my "Second Memoir upon Quantics," Phil. Trans. vol. cxlvi. (1856).

VII. "Tables of the Sturmian Functions for Equations of the Second, Third, Fourth and Fifth Degrees." By ARTHUR CAYLEY, Esq., F.R.S. Received December 18, 1856.

(Abstract.)

The general expressions for the Sturmian functions in the form of determinants, are at once deducible from the researches of Professor Sylvester in his early papers on the subject in the Philosophical Magazine,' and in giving these expressions in the memoir "Nouvelles Recherches sur les Fonctions de M. Sturm," Liouville, t. xiii. p. 269 (1848), I was wrong in claiming for them any novelty. The expressions in the last-mentioned memoir admit of a modification by which their form is rendered somewhat more elegant; I propose, on the present occasion, merely to give this modified form of the general expression, and to give the developed expressions of the functions. in question for equations of the degrees, two, three, four and five.

January 15, 1857.

The LORD WROTTESLEY, President, in the Chair.

The following communications were read :—

I. "Photo-chemical Researches. Part II.

Phenomena of

Photo-chemical Induction." By Prof. BUNSEN of Heidelberg, and HENRY ENFIELD ROSCOE, B.A., Ph.D. Communicated by Prof. STOKES, Sec. R.S. Received November 27, 1856.

(Abstract.)

Chemical affinity, or the force which regulates the chemical combination of two bodies, is like all other forces, a certain definite quantity. Hence it is erroneous to say, that under different circumstances the same body can possess different affinities; more correctly we should say, that in the one case the bodies are able to follow the chemical attraction of their molecules, whilst in another case opposing forces render this combination impossible. These opposing forces may be considered as resistances similar to those exerted in the passage of electricity through conductors, in the distribution of magnetism in steel, and in the conduction of heat. We overcome these resistances when by agitation we increase the formation of a precipitate, or by insolation effect a decomposition.

We call the act by which these resistances to combination are lessened, and the formation of a chemical compound promoted, "chemical induction;" and we specify this as photo-chemical, thermochemical, electro-chemical, or idio-chemical, according as light, heat, electricity, or pure chemical action is the force which promotes the combination.

The phenomena of photo-chemical induction are particularly interesting, as affording starting-points from which we may gain a knowledge of this mode of action of affinity.

That on exposing a mixture of chlorine and hydrogen to the light the action does not commence to its full extent at once, was observed