QUEENSLAND 
PHILOSOPHICAL SOCIETY 
(From the Queensland Guardian , May 10, 1865.) 
At the monthly meeting of the above society 
held on Monday, May 8, Chief Justice Cockle, 
President of the society, read the following 
paper : — 
His Honor said : — The theory of Leibnitz 
tends to bring into conformity the notation of 
algebra and of the differential calculus. The 
substantial connection between the two sciences 
is more clearly manifested in the solution of 
linear differential equations with constant 
coefficients, and in that integral, the discovery 
of which, by Euler, was the first step towards 
the foundation of the theory of elliptic func- 
tions. The approach of the two sciences is 
further traceable in the researches of Abel and of 
Jacobi, of G-alois, of Hermite, and of Kronecker. 
Some time before leaving England, I showed 
that the solution of an algebraical equation of 
any degree (or, what for the moment I shall 
treat as a synonym, of any order), may be 
made to depend upon that of a linear differen. 
tial equation, of an order inferior by unity to 
that of the algebraical equation. These dif- 
feren ial equations are now known by the 
name “ differential resolvents,” and I certainly 
thought that they would be found to yield to 
some known process of solution, as, indeed, the 
resolvent of the imperfect cubic did. But a 
distinguished writer on the subject of algebrai- 
cal equations, Mr. Harley, (now) F.R.S., who 
took a great interest in the subject, and calcu- 
lated certain higher differential resolvents, found 
that in general (and this remark applies to 
that of the imperfect biquadratic) they were 
not directly soluble by any known method, and 
he discovered that the resolvents of certain 
trinomial equations take peculiar forms which 
may be regarded as additions to the primary 
forms of Boole. I am now, I believe, able to 
announce the following results in the theory of 
linear differential equations. (1.) I have been 
led to an a priori demonstration that the 
