202 
Mr. Harley also laid before the Society the following 
communication from Mr. Cockle, dated Temple, February, 
1862. 
The theorem which I gave in my first paper, “ On Tran- 
scendental and Algebraic ■ Solution,” in the Philosophical 
Magazine for May, 1861, viz., that the solution of an 
algebraic equation of the w-th degree, whereof the coefficients 
are functions of one parameter, depends upon that of a linear 
differential equation of the (n — 1) th order, may be applied 
to any algebraic equation whatever, without any preliminary 
modification of its coefficients. And the simplest, or nearly 
the simplest, form of the general process indicated in my 
second paper (“ Supplementary Paper,” Philosophical Maga- 
zine for February, 1862) with the above title is, as I have 
pointed out to Mr. Harley, obtained by treating all the 
coefficients as constant and multiplying the last into a para- 
meter x, which is to be treated as the independent variable. 
Thus, given the quadratic 
0 , 
we deduce, successively, from 
y*+1n/+cx=0, 
the following relations : — 
dy 
dy 
<1 yip + h 'dP + *- 0 ’ 
dy _ 
dx 
dy 
dx 4 cx — b 
2 y+b 
2 cy 
(2 y+b) 
y — C V 4 cx — b 2 — -~- 
4 cx — b~ * 
be 
4 cx — b 2 ' 
b_ 
2 
In this expression x is to be made equal to unity, and the 
arbitrary constant C determined by substituting for y in the 
given quadratic, or by processes which will, I hope, soon be 
explained by Mr. Harley. For the general sextic I should 
be disposed to deal with the form (attainable by Mr. Jcrrard’s 
process, by vanishing groups, or by Mr. Sylvester’s method), 
