Easton and Amos, who procured for me twenty-two solid pillars, each 10 feet long and 2 inches diameter, cast out of eleven kinds of iron (nine simple irons and two mixtures). The pillars were all from the same model, and were cast vertically in dry sand, and turned flat at the ends, as the hollow ones had been; two being cast from the same kind of iron in each case. The simple unmixed irons tried were as below, and all of No. 1.

The mean strength of the pillars from the irons above varies from 20.05 to 29.50 tons; or as 2 to 3 nearly.

The pillars formed of mixed irons were found to be weaker than the three strongest of the unmixed series.

From many experiments, it was shown that the weight which would crush the pillars, if they were very short, would vary as 5 to 9 nearly.

The pillars in general were broken of four different lengths, 10 feet, 7 feet 6 inches, 6 feet 3 inches, and 5 feet, the ends of all being turned flat, and perpendicular to the axis. It was found that when the length was the same, the strength varied as the 3.5 power of the diameter; and when the diameter was the same and the length varied, the strength was inversely as the 1.63 power of the length. Both of these were obtained from the mean results of many experiments. The formula for the strength of a solid pillar would therefore be

where w is the breaking weight, d the diameter in inches, the length

in feet, and m a weight which varied from 49.94 tons in the strongest iron we tried, to 33.60 tons in the weakest.

The ultimate decrement of length, in pillars of various lengths but of the same diameter, varies inversely as the length nearly. Thus the ultimate decrements of pillars 10 feet, 7 feet 6 inches, 6 feet 3 inches, and 5 feet, vary as 2, 3, 3 and 4 nearly, according to the experiments, from which it appeared that the mean decrement of a 10-feet pillar was 176 inch.

Irregularity in Cast Iron.

The formulæ arrived at in this paper are on the supposition that the iron of which the pillars are composed is uniform throughout the whole section in every part; but this was not strictly the case in any of the solid pillars experimented upon. They were always found to be softer in the centre than in other parts. To ascertain the difference of strength in the sections of the pillars used, small cylinders inch diameter and 1 inch high, were cut from the centre, and from the part between the centre and the circumference, and there was always found to be a difference in the crushing strength of the metal from the two parts, amounting perhaps to about one-sixth. The thin rings of hollow cylinders resisted in a much higher degree than the iron from solid cylinders. As an example, the central part of a solid cylinder of Low Moor iron No. 2, was crushed with 29.65 tons per square inch, and the part nearer to the circumference required 34.59 tons per square inch; cylinders out of a thin shell half an inch thick, of the same iron, required 39.06 tons per square inch; and other cylinders from still thinner shells of the same metal, required 50 tons per square inch, or upwards, to crush them.

As these variations in cast iron have been little inquired into, except by myself, and have never, so far as I know, been subjected to computation, I have bestowed considerable trouble upon the matter, in an experimental point of view, and endeavoured to introduce into the formulæ previously given, changes which will in some degree include the irregularities observed.

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, 1856.

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