of Edinburgh, Session 1871 - 72 . 
591 
or. 
(i ( i+i ) -.o- 1) ) (l -^* 1 ^+£(a-^)=o ( 4 ). 
3. Let Q j be any one of the values of Q above defined, then 
r s d s Q; d s Q 9 - s d s Q { d s ~ l Q,- /% d s ~ l Q; d / 1 c^QA 
— (1 — /X 2 ) S d S ~ l 0? 4- (i 4- o\ (i — s 4- 1 ) C(\ — »2\s—1 d S ~ 1 ^ d S ~ l Qi dr 
~ ( 1 4 ^r + (i + s)(i 5 + 1 V (1 M 
Hence, integrating between the limits nFl of /x, we have 
/ < i -^ , ^*= (< +* )(< -* +i >/ (6) - 
+ i +i 
Applying the reduction s times, w r e evidently obtain 
—l _____ ,. —i 
f a-/**)' 
Ai 
s cl s Qi cl s Qj 7 |Hyf 
djx s cI[jl s 
djx — tt 
i — s 
j QiQjd/x (6). 
Hi 
4. To find the value of the integral on the right, note that 
QiQj is the co-efficient of lc li j in the expansion of 
1 
Now 
i 
(1 - 2ph +h 2 )i (1 - 2ij.h' + h' a )i 
d.u 
a /(1 +P -2 V) (1 + A ' 2 - 2 A » 
1 , V 2 h 
I no* - 
, / “7T7T” — 1 
1 + A 2 
7 
1 + A ' 2 
2 A ' 
v'AA' 
'l + / i 2 . „ , /1 + A ' 2 
2/i 
+ 1 + 
j 
2/i' 
1 . \//i' (1 - h) + */h (1 - 7i') 
loo 1 - 
*/hh' °Vl'(l+l) + ■x/A (1 + A') 
1 l-v'AA 7 
,== log. -^ 7 = 
a/AA' ° 1 + a/AA' 
