183 
and which also enables us with ease to calculate the resolvent 
for the cubic (n = 8), viz. 
+»=°-( a ) 
Results (1) and (2) were first calculated by a different process 
by Mr. Cockle. The following are now published for the 
first time. For the biquadratic ( n = 4) the differential 
resolvent is 
2 3 (1 — x 3 )^— 2 4 -3V^ — 2-43*^ + 5y = 0--. 
car cur r/x 
And for the quintic ( n — 5) the resolvent is 
2 . |-3 2 . 5 3 .13x 2 ^ 
v 7 c/x 4 ete 8 dx 4 
— 3-5 3 .l7.r^ + 3.7-lly = 0...(4) 
ax 
(3) 
The Boolian form of (3) is 
7w~ 10 n 
13\ 
(“- )( d -t)( d -t 
/ 30 
D(L) — 1)(D — 2) 
e ol, y = 0...(5) 
d 9 
where D is the differential symbol — , and e =x. 
I notice also the following remarkable relations among the 
differential coefficients for the general case, viz., 
£=*-&)’■ 
(a— »sr-+c«— *) } [%\\ 
JfL-y"-’ j — (2n — 1 ) (3 re — 1) are/" -1 
dx l n — L ( 
+ 3(n— l)(» + 2)(2»+l)y 
-(2re-l)(3re 2 +3re-8)*J j^J • 
The investigations on this subject, which I intend to 
publish shortly, embrace a curious and anomalous result in 
