[710 
710] 
ON A DIFFERENTIAL EQUATION. 
19 
irithm of a 
its of which 
the integral 
2, where fl 
r , D are of 
,1 is 
are rational, 
uation 
I had previously obtained the solution 
/1 — \Za?y 
Vl+^V ’ 
and I wish to show that this is, in fact, the particular integral belonging to the value 
0=1 of the constant of integration : for this purpose I proceed to rationalise the general 
integral as regards 
Writing for a moment 
P = 0+ 1)0 2 - 34*+1), 
Q = (z 2 + 14;z + l)Vp + 14^+ 1, 
JR, = MOz (z — l) 2 , 
where 
M — C + 1) ( x ‘ ~ 34a? 4-1) 4- (a? 2 4- 14# + 1) Va? 2 4- 14a? + 1 
Va? (a? — I) 2 
the integral is P + Q + JR = O', or rationalising, it is 
(P 2 - QJ - 2P 2 (P 2 + Q 2 ) + P 4 = 0; 
we have 
P 3 = ( 1, -66, 1023, 2180, 1023, - 66, 1\z, l) 6 , 
Q 2 = (l, 42, 591, 2828, 591, 42, 1 \z, l) 6 , 
onr] thpiipp 
P 2 -Q 2 = (0, -108, 432, -648, 432, -108, 0\z, l) 6 , 
= - 108^(^ — l) 4 ; 
P 2 4- Q 2 = 2 (1, -12, 807, 2504, 807, - 12, 1\z, l) 6 . 
Writing the equation in the form 
i(P’ + <?) - 1 y+=0, 
it thus becomes 
(1, -12, 807, 2504, 807, -12, 1\z, l) s -z(z-l) 4 jilP + = 0, 
where M has its above-mentioned value; and if we now assume 0=1, then 
(a? + 1) (a? 2 — 34a? 4-1) 4- (a? 2 + 14a? + l)Va? 2 + 14a? + 1 
= Vx (a? — l) 2 ’ 
108 (a? 4-1) (a? 2 — 34a? 4-1) — (a? 2 4- 14a? 4-1) Va? 2 4-14« 4- T 
M Va?(a?—I) 2 
and thence 
.ZIP 2 4- 
(108) 2 
if 2 ’ 
M - 
108V 
M) 
+ 216, =4 (« + iry-34r + iy + 216| 
00 \X JL ) 
a? (a? — 1)' 
.(1, -12, 807, 2504, 807, -12, l\x, l) 6 : 
3—2