418 THE THEORY OF TERRESTRIAL MAGNETISM. 



10. We have seen above that 



) 



1 J)m p (n + m)(w + m- 

 3 2-l2n+l 



Multiply by (lfjf) m D m P ni and integrate between limits, supposing 

 that ! and n are both odd or both even. Then 



-l -l 



which vanishes except when n 1 = ?i, or when ?tj = M 2, or when n t = 

 When n t = n 2 



2/i 3 (nm 2)l (2n3)(2nI)(2n+l)(n 1 i 



nd I 1 (\ u?\H m H m du.- (n + m)l(n-m)l 



JV 2n+l r3;5...(2n-l)l.3.5...(2-l)' 



When n^ = n + 2 we get 



2(n, 



__ 

 (2n + l)"(2n + 3) (2n + 5) (n - m) ! ~ (2n + 1 ) (2n + 3) (2n + 5) (n - m) ! ' 



nd 1 



-i ' 2w+5) l35(2+3) 1-3.5 ... 



Hence, when n, = w 2, 



^H m H m diL 2 (n + w)!(n-m)l . 



/i .tf H J tf ni d /4 _ 2 __ ^ 3 5 (2n _ l)}8 ............... (9), 



and when n, = n + 2 



2 (ri, + m) ! (w, TO) ! 



= 



In all other cases this integral vanishes. 



