Mutual Induction. 371 
( 7 — | = coefficient o£ k n in the expansion of 
2r+l 
3.5...(2^-l)F(l+^) 
or 
D; ( 0) = 3.5..X2,-l)|(^p±p^(-l) ,i ?| 
1 (;. + «-!)! (-1) 
n—r 
~2~ 
(-1) 
f'- 1 /»• + » — 2\, / n — y \ 
V 2 7 I i ;• 
i. e. 
J j w ^ 2 /t_2 Lm — 2 / w + m — 3 \ T /?i — ?/? 4- 1\ 
(n 4- m — 1) (?z — m 4- 2) m — 3 (n + m — 4) ! 
«i — 2 ' m — 4 n + m — 5. w — ra4-3. 
+ 2 ' ' 2 * ! 
(n+m-2)! f, ,. , m-3 r . , 
I //I— 1)2™ 2 "^_ ^y '" "» ' f V_ no ^t 
2 " " 2 ' + "l 
-C(saj) ■ ' ' 'J 
where 'ii — ??i + l must be even ; otherwise the integral is equal 
to zero. 
3 even 
»-i 1 ft* 
= (~ 1 ) 2 2^T /n _i\ i 2 if wis odd. 
KV 1 )-'} 
Hence, ultimately, 
4 • A ; o\2p+l)\2p + 2y dr ' dr' 2 4 p(;,'.)^ 
ifiA^y ( n - m > ! (^-i)(^ 2 -i) <*Q» rfQT «,/. 
where m must be odd, and n must be even. 
4. If the ellipses are parallel, with the line joining the 
centres perpendicular to their planes, the same investigation 
will apply, only the limits of integration for <f), <j>\ /u, fx' will 
be different. 
