44] 
CONNECTED WITH THE THEORY OF ATTRACTIONS. 
287 
The only practicable case is that of q' = — q, for which 
U = 
Tr^ n I 1 (4-n) 
s i n q l 
F (%n + q) F (\n — q)J 0 (v‘ 2 s 2 +js + u 2 )i n 
Consider the more general expression 
.(8). 
0 
’rr* *(**+* + * 
ds 
•(9); 
by writing 
2 u*Js = \/(s' + 4<uv) 4 \/s', 
the upper sign from s = co to s = |, and the lower one from s = ~ to 5 = 0, it is easy to 
derive 
0 = (2v) 2q 
{C(s + 4mv) + \/s} 23 + {\/(s + 4uv) — V«} -23 
\/s \/(s 4- 4uv) 
Now, by a formula which will presently be demonstrated, 
l 
2\]it 
(f> (s + j + 2uv) ds (10). 
W( s + + Vs]~ 23 + + 4uv) - \/s\-- q _ 
\/s V(s 4- 4uv) 6 S 
6 q \ s i q (s + 4tuv) i q e 9s ds (11)5 
whence 
f°° {\/(s + 4uv) + Vs} -23 + {V(s + 4uv) — Vs} -23 ^ 7 
Jo y/sV(s + 4uv) * s - ds 
2 \Jtt 
s~i~ q (s 4 
ds 
Fs . ds (12). 
re*-?);« 
Thus, by merely changing the function, 
2234-1^2?^. r® / ¿7\-3 
0 = T(f - r/)" J 0 S ~^ 9 ( S + * uv )~ l ~ q { ~ ds) $ ( s + j + 2uv ) ds (13); 
and hence in the particular case in question 
2 2 ?+l tfq TfJ (n+l) r® 
^ = r(y-g)'r(pTg) J o S ~ } ~ q ^ + ( 5 +i + 2uv)~i n+q ds (14), 
by means of the formula 
d\~ q , . T(iw- 9) 
(-*")’ < s+ “>■*” - (s + 
But as there may be some doubt about this formula, which is not exactly equivalent 
either to Liouville’s or Peacock’s expression for the general differential coefficient of a