QUEENSLAND 
PHILOSOPHICAL SOCI E T Y 
(From tho Queensland Guardian , August 7, 1866.' 
At a meeting held on Monday, July 30, 
1866, tlie President of tlio Queensland Philo- 
sophical Society, Chief Justice Cockle, F.R,S.» 
read the following paper — 
ON THE INVERSE PROBLEM OP CORE SOLVENTS. 
Inverse problems, as is well known, present 
greater difficulties than direct ones. For in- 
stance, while it is easy to square a number, it 
is not so easy to extract its square root. More- 
over, there are cases in which it is impossible 
to obtain a finite solution of an inverse 
problem. The solution of a quintie is usually 
considered to be such a case. In the 
theory of co-resolvents it is comparatively 
easy to pass from the algebraical to tho 
differential resolvent, but the converse does not 
hold. The finite integration of the linear 
differential resolvent of a given_ algebraical 
equation would perhaps be a step "towards the 
general solution of the inverse problem. But 
that integration has not yet been effected ex- 
cept in two or three special cases, and the de- 
finite integrals of Boole have not, that I am 
aware of, been converted into indefinite ones. 
In order to take the step above pointed to, it 
[ seems to me necessary to have recourse to a 
non-linear differential resolvent, to be con- 
structed as follows The elements of the final 
non-linear are three. The first is (1) 
tho second differential co-efficient of the de- 
pendent variable ; the second is (2) 
the first differential co-efficient of that variable; 
the third is (3) the square of the second element 
divided by the dependent variable itself. The 
sinister of the non-linear resolvent is constituted 
by the six homogeneous quadratic products of 
the three elements, and is the sum of those six 
products, each multiplied into an indeterminate 
or conditional multiplier. Each element, and 
each produot, is, as we know by the theory of 
coresolvents, in general capable of being ex- 
pressed as a rational and integral function, 
of tho dependent variable, of a degree 
less by one than that of the given 
algebraic equation. Suppose this last equa- 
tion to be a quartic. Then each product and 
consequently, tho dexter of tho non-linear re- 
solvent can be expressed as a cubic function of 
the dependent variable. Let the dexter of the 
non-linear be reduced to zero by causing the 
