414 THE THEORY OF TERRESTRIAL MAGNETISM. 



+ r (2n - 2r + 1) (2n - 4r + 1) (_ + &c., 

 hence -^ = (2w - 1) (2n - 3) QT-I + 2(2n- 3) (2n - 7) QT-l + &c. 



1 /A 



+ r (2n - 2r + 1) (2w - 4r + 

 From which we obtain 



#" .-7/- + 2 /2n S^ (n-m)(n-m-l) gm . 2 

 1^ - 5) (2^lH2w - 3) (2w - 5) ^- 4 H 



, (n m) (n ml}...(n 

 + ,- (2n - 2r + 1) 



Multiply the value of (1 - //) 2 Z> m+2 P n obtained above by (l -//) 2 Z) m P n _ 8 

 and integrate from //,= ! to /u, = l, then 

 i 



i 



-r(2n-2r+l)(an-4r+l 



7 



-i 

 2 



4r + 



Putting n 2r = n lt we get 



f ' (1 - nTD^Pn DP n d . (n - n,) (n + n, + 1) g 

 ^1 \'''i 



We have seen that 



' Hm -i , o / }} { 1 H-m-2, 

 " " } - "- 2 



(2w-l)(2n-3)(2n-5) - 4 > 



(n-m)(n-m-l)...(n-m-2r + S) 



(2n-l)(2n-3)...(2n-4r + 3) ' a **' t ' * 



