THE THEORY OF TERRESTRIAL MAGNETISM. 413 



Now multiply by H^dp. and integrate from /i= 1 to /t=l, then 



*, ,-, , _ (n-w-l)(n-m-2)...(tt-m 

 ' "-*'" (2-l)(2n-8)...(2n-4r 



( 



(2n - 1) (2n - 3) .. . (2n - 4r '+3)"" 



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

 {1 .3.5... (2w-4r-l)} s (2w-4r+ 1) 



- 4m ( - m - 1 ) I (n + m - 2r-^ )| 



1.3.5 ...(2-l)1.3.5 ...(2n-4r-l) " 



Putting n 2r = n l , and recollecting that n and Wj are both even or 

 both odd, and that when one of them is even and the other odd the 

 integral evidently vanishes, we have 



u . ( n -m-l)\(^ + m-l)l 



li 



where n, is less than n. 



Writing n^ for n in the above equation before integration and multi- 

 plying by H~ l dp. and then integrating we get 



since all the quantities n lt t^ 2, &c. are less than n- and so all the 

 terms separately vanish. 



Again multiplying the above equation before integration by H~ 1 d/j, we get 



(n-m+l)\(n + m-l)l 

 ~ 2 " 



7. We have seen above that 



+ 3 (2n - 5) (2n - 1 1) P u _. + &c. + r (2n - 2r + I ) (2n - 4- + 1 ) P a _ w + &c. 



Hence D m+ *P n = (2n- 1) (2n- 3) D m P n _, + 2 (2n - 3) (2n- 7) D m P n _ 4 



+ r(2n-2r+\)(2n-4r+l)D m P n . a _ r 



