THE THEORY OF TERRESTRIAL MAGNETISM. 415 



and 



#L . _ - + o (2n - 3} (n-m)(n-m-l) Hm , 2 

 l-~ 6) 2n-l2n-32n-5" n -^ 



. (n m) (n m 1} ... (n m 2r + 3) TT 

 + r(2n-2r + l) ^^^n _ 8 { ../ (2n _ 4r + 3) ^ + Ac. 



Multiply this last equation by Hdp. and integrate, then 



T> , 1 w--- (n-m)! (n, + w-2)! 



" - 



/"i (J:?'") 2 

 Wi being less than n, hence the value of the above definite integral I ^ -^dfj. 



may be found by substituting n 2, n 4, &c. successively for Wj in this 

 equation. 



8. We will now find a formula of reduction for P (1 - ^) m+1 (D m P n Ydfji. 



J-i 



We have Been that 



jjl -') 



and P .- 



also from a previous paper (see Vol. I. p. 488) we have 



3 (ro+l)(n + 2) p n(n+l) 3 n(n-l) p 



" ' 22w+l2n + 3 2n- 1 2n + 3 " 2 2w- 1 2w+ 1 n ~ 



he 



2 n(n + l) 

 n+2 + 



3J ~(2n+l)(2n+3) n+2 3 (2w- 



-) p 



"- 2> 



fu' DP - p _ 2 



* 1 - + 3) " +2 (2n-/i 



w(n 1) 



(2n-l)(2n + 



