240 
y — A 
could not by any determination of C give each of the roots 
y—y x , y=y»- The differential equation must have a term 
independent of y, and the solution is 
y=yi+C(i— yO 
which in virtue of y x -\-y 2 — 2, is 
=y 2 +(2— C) (1— y 2 ) 
and gives y x or y , as C=0 or C=2.” 
Mr. W. H. L. Russell, of Shepperton, Chertsey, has 
favoured me with the following elegant solution of the 
quartic resolvent obtained by the aid of definite integrals. 
( 5 2 1 /17 5wl4 2\y 11 lx N 
\T iyVT 4/VT 4 / 6 
-f-Bx 
+CV 
=A 
1 + 
1 + 
4 4 
3-2-1 
9 6 3 
4*4*4 3 
4-3-2 
13 10 7 
x 3 + 
x 3 + 
(6 -3) (5 • 2) 4 • 1) 
>21 9\ / 18 6\ /\5 3\ 
\7 4/ \T 4/ '-T 4' 
■ X 6 -f . 
4 4 4 3 . 
1 + “e — a — o' x + 
(7 • 4) (6 -3) (5 • 2) 
r 25 
vT 
x c -f 
S25 13XX22 10-v W9 7^ 
4 y V i TyvT'Iy 
5 4 3 
1048576x 3 
(8-5) 
UL - 5 . 
X V ’+ 
819tT 
(7 -4) (6 • 3) 
'fa? f'f'f' V* z 2 to 1 (1 — vfe (1—2)^ (1— wf* dv dw dz 
i 0 "o 0 1 VZWX 3 
4-Bx f ’ f ' f' V ' z ~ w * — w )‘ (1 — ~) J (1 — w) 1 df dz dw 
0 0 0 i— vzwx 3 
SL 4 . 3 . 
,4 -2 
. p, 2 f'f'f' * 2 ‘ (1— t') 1 (1—2)' (1— ‘ (72 (/«> 
^ ^ ^ ^ 1— VZU'X 3 
0 0 0 
In this solution a: is assumed less than unity. An inter- 
esting paper by Mr. Russell, on the Theory of Definite 
Integrals, will be found in the “ Philosophical Transactions” 
for 1854, pp. 157-178. 
Dr. Boole lias also pointed out to me a method of solving 
the quartic resolvent, which does not, however, essentially 
differ from the above. And Mr. Russell remarks that a 
