13 
calculated, and I believe I had also communicated to Mr. 
Cockle, the Boolian forms of the resolvents for the cases 
n — 2 to n — 5, both inclusive; and these suggested to me 
the general form (A). (See Proceedings , vol. II., pp. 
199—201, and pp. 237—241.) 
The singular simplicity of these results for the form (I) 
had an effect inducing me to consider the corresponding 
form. 
y n — ny n ~ l -\-(n — l).r=0 . . . (II) 
to which also any algebraic equation of the «fh degree, 
n being not greater than 5, can, as Mr. Jerrard has shown, 
be reduced by means of equations of inferior degrees ; and by 
induction I was led to the following general expression for 
its resolvent, viz. 
n"~\(n — 1)D]’ !_1 y — {n — 1)(»D — n— 1)[«D — 2]' !_2 e^ y 
e 6 . . . (C.) 
or, what is the same thing, 
—\n — 1 ]’ 1_1 x . . . (D) 
The particular cases on Avhich these inductions were 
founded are given in the present Memoir. 
Every differential resolvent may be regarded under two 
distinct aspects. It may be considered either, first, as giving 
in its complete integration the solution of the algebraic 
equation from which it has been derived ; or, secondly, as 
itself solvable by means of that equation. In the first aspect 
I have considered the differential equation (A) in a Paper 
entitled <f On the Theory of the Transcendental Solution 
