of Edinburgh, Session 1871 - 72 . 
595 
whence, 
7 l-/x 2 h 2 d fl -p*\* _ 
y-H + h 2 + 1 . 2 ^V i 2 / + &c, > 
and therefore, 
1 -^-1 | 7. - 
V 1 + 2 /x/i + /i 2 d/*- dfx V 2 
d /1 -m 2 \ A 2 d 2 /1 -m 2 \ 2 d 
(_ — 2~) + L2 cp* (gl - j + &0 -’ 
which shows that 
Q; 
= (-)•( 
/ "LY f—>' 
djx 
( 21 ), 
and suggests obvious simplifications of preceding results, e.g. } 
6s =( - § 8) ( - y i+s{i - 
&c., &c., 
11. The complete integral of 
.(i+Dfe + Ko-, 1 )*,) -0 . ( 3 ) 
may easily be found, since a particular integral is known. Let it 
be MQ*, where M is a function of p. Then (3) gives at once 
( 
- 2 ,Q i+ 2(l_^)f)f + (l-^ Qi ^ = 0 
or, 
- 2 /x 2 dQi 
+ nT 1 + dW 
whence 
1 — /x 2 Q i djx 
dM 
d[x 
c 
d 2 M 
d/x 2 
0 
d[x (1 — fx 2 )Q l 2 
Thus the complete integral is 
dfx 
cq / 
(1 - fx 2 )Q 4 
• • 
12. Let us now suppose 
( 22 ). 
