~ 
152 Royal Society. 
3 inches, and 5 feet, vary as 2, 3, 34 and 4 nearly, according to the 
experiments, from which it appeared that the mean decrement of 
a 10-feet pillar was *176 inch. 
Trregularity in Cast Iron. 
The formule 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 cylin- 
ders 3 inch in diameter and 15 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 m 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 re- 
quired 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, re- 
quired 50 tons per square inch, or upwards, to crush them. 
‘As these variations in cast iron have been little inquired into, ex- 
cept 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 intro- 
duce into the formulz previously given, changes which will in some 
degree include the irregularities observed. 
«Memoir on the Symmetric Functions of the Roots of certain 
Systems of two Equations.” By Arthur Cayley, Esq., F.R.S. 
The author defines the term roots as applied to a system of n—1 
equations ¢=0, =0, &c., where ¢, W, &c., are quantics (7. e. rational 
and integral homogeneous functions) of the n variables (a, y; ¢, . -) 
and the terms symmetric functions and fundamental symmetric func- 
tions of the roots of such a system; and he explains the process, 
given in Professor Schlifle’s memoir, ‘“‘ Ueber die Resultante emes 
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 sy- 
stem of n equations is made to depend upon the very simple question 
of the determination of the resultant of a system of ” 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 equa- 
tion; 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 
