72 
A Paper was read, “ On the Solution of the Differential 
Resolvent,” by W. H. L. Russell, A.B., communicated by 
the Rev. Robert Harley, F.R.A.S. 
“ In the course of last year I was induced, at the request 
of Mr. Harley, to consider the very interesting differential 
equation which he has denominated the ‘ Differential 
Resolvent.’ I obtained the solution of the quartic resolvent 
by series, which I summed by means of a triple integral. 
But Professor Boole intimated that he had discovered a 
process of transformation by which the quartic could be 
resolved by a single integral. This led me to examine my 
own series, and I find that the series representing the solution 
of the quartic could be summed by means of a single integral. 
I have since discovered that the general resolvent can be 
solved by means of a single integral. To effect this is the 
object of the present Paper.” 
The Author first deals with the resolvent of the quartic 
equation, viz. : — 
f — 4 y + 3x, 
( d — 1 )( p - t )( d - t ) m „ 
y e ?/“0 
J D(D — 1)(D — 2) J 
where ® — x; and he obtains the solution 
rr ( 5a 0 (3xf) 
n COS^0 { cos-— cos *0 cos-- ■ - > 
tfdxxf\ d0 
I 
3 
2 1 j 
/ J 3*2/ \ ^ 
1 — cos 4 0cos20^— j + cos p 0 
2 s 
C being a certain irrelevant function. 
He next discusses the resolvent of the equation 
y n — — l)x= 0 , 
viz. : — 
where 
0(D) 
( i\° 
y — 0 (D)e^ M ^ y=o, 
D(D— -lj(D — 2 )...(D — h + 2). 
