of Edinburgh, Session 1871-72. 
595 
whence, 
7 1 -g 2 h 2 d ( 1-/x 2 Y , 
y = H+h— 2~ + §— ) + &c -> 
and therefore, 
1 _ dy _ d_ df / l-^ Y 
v l + 2/J> + A a ~<^~ 2 J + l-2 d^ 2 \ 2 y +cC,> 
which shows that 
■ • ( 21 >’ 
and suggests obvious simplifications of preceding results, e.g., 
c • - - (by § 8 ) ( - ) i+s ( i - "■ 
&c., &c., 
[ t- s \c?jtc / \ 2 / ’ 
11. The complete integral of 
^DQ. + Ka-^f) =° . (3) 
may easily be found, since a particular integral is known. Let it 
be MQj, where M is a function of Then (3) gives at once 
(- VQ>+ 2 cw*>f )f + a - = o, 
- 2a , 2 dQi -4- dm 
l-ju 2 + Qi d/l + ~ dy* ~ °> 
d/ji 
whence 
dM 
G 
dfx (1 - p 2 )Q,i 2 
Thus the complete integral is 
ca fiF$W 
12. Let us now suppose 
Si = P,Qi • 
(22). 
(23), 
