of Edinburgh, Session 1871-72. 
591 
3. Let Q j be any one of the values of Q above defined, then 
Hence, integrating between the limits ”Fl of p, we have 
f 4.= (.•+«) (i-.+ijy (5 ). 
+ i +i 
Applying the reduction s times, we evidently obtain 
—i „„ —i 
J O-/ 4 ) dll ,, 
d s Qi d s Qj 
~dfd 
|t + s 
^ tl 1737 
J Qi 0/ *7/^ 
( 6 ). 
4. To find the value of the integral on the right, note that 
QiQj is the co-efficient of A*A^' in the expansion of 
Now 
(1 - 2 fxh +h 2 f (1 - 2^' + A' 2 )* 
dp 
a/(1 + A 2 - 2 V) (1 + h' 2 - 2 A» 
j 
+i 
7IF los - 
/ 1+A 2 
2 li 
1 + 
l + A /; 
2A' 
j 
1 + A 2 
~2/T 
+ 1 + 
/1 + A' : 
V 24' 
+ 1 
1 ^ a/ A'(1-A) + \A(1 -AQ 
a/ AA' ° \/A' (1 + A) + a/A (1 + A') 
1 1 - a/ AA' 
a/ AA' ° 1 + a/ AA' 
