146 
S. Equation (6) is made linear of the second order by 
the assumption of 
y= 
8a s + 27x\ du 
dx 1 
dx 
udx 
(ii) 
dhi 
dx 2 
Substitute the above value of y in (6) and we obtain 
dxi 
dx 
. K dx i d 2 x i 
^ dx 2 f 
8a 3 + 27^ ( <7# 
dx 
+ 18a 3 
/ ^ \ 2 
(^8a 8 + 27^ 3 J w = 0 (1 2 ) 
The general integral of (12) is, therefore, the value of (u) 
determined from (11) wherein y takes the value given in 
equation (10). 
4. The following particular case is of some interest in the 
theory of elliptic functions. 
Put, X\ — -4^# + ^, and a = 6. 
Substitute these values in equations (1) and (6); then 
they become as follows : 
'/-Qf-i(x + l)y-3 = 0 (13) 
3(1 -»■ + (* + i)y + 3 = 0 (14> 
The various steps in the substitution and reduction, being 
simple, are omitted. 
Equation (14) has been considered by Professor Cayley in 
the Messenger of Mathematics, Vol. IV., pages 69, 110, and 
Elliptic Functions, page 248 ; and also by Mr. Hart, in the 
Messenger of Mathematics, Vol. IV., page 125. 
The general solution of (14) is given by the substitution 
of Xi = - 4^ + and, <z=6 in equation (10) and is as 
follows : 
/ x 2 - l\i 2 f y\c 
(—) ° xp ' V T- 
Vidx 
x 2 
c + 
lf\( 1 y exp. 2 f^S— l dx 
3 ; Iw-W 3 J ij x clx I 
y=y i+ 
(15) 
