430 THE THEORY OF TERRESTRIAL MAGNETISM. 



and 



P r; + x: = ( i - ^ (n - m) {x'- 1 "- 1 - &c. 



i Y nl(n-m-l)\(2n-2r)\ 



' 

 hence 



tn-l 



/* F,;- + X? = ( 1 - p.*) " (n - m) ( 1 - /1 s ) " - &c. 



( _ 1 



1 



J ' 



Multiplying the last two factors together, the coefficient of ^"-" i - :r+1 i n 

 the general term of the product is 



n 



(2n-2r)! 



2n! \(?i-m-2r-l)! r! (w-r)! 



(2n-2r+2)! 



l 



(?i-m-2r+l)! (r-1)! (n-r+1)! 



- 



2n\ rl(n-r)l(n-m- 2r + 1)1 



x {(n -m- 2r) ( - m - 2r + 1 ) + 2r (2n - 2r + 1)} ; 



-. o _ rT . 



2 2n ! r ! (n r) ! (n m 2r + 1 ) ' 

 x [(n - m - 2r) (n - m - 2r + 1) + 2r (2?i - 2r + 1 ) + (n + in) (n-m+l)] ju ,"- m - sr + 1 1 . 



The quantity in square brackets becomes 



(n -m + 1) (2n - 2r) + 2r(n+m+l)2x[n(nm+l) + 2mr\. 

 Hence 



X'"- -a-u^"|nu"-" +1 -&c f( I *!(n-m)!(2n-2r)! 



^ } '* ) 



x [w (n - Tit + 1 ) + 2mr] /i "-'"- 2 '-+ 1 + & c . 1 , 



, 



!r! (?i r)! (n m 2r)l 



^^ 



2nlr\(n r)\(n m 2r) ! r } 



