THE THEORY OF TERRESTRIAL MAGNETISM. 417 



- mw + 2 - 



(* - n n P - __ _ 



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



(n + m)(ro-l+ 

 (2n-l)(2n+l) 



Now multiply by ( 1 /jf)'" D m P n and integrate between limits, then 



-* (^{(ti-s) I'/ 1 -/rywv* 



n 2 + n + m 2 -! (n + m)! 



- - ' 



9. From the above equation we derive the following definite integrals : 



r, \/rrmy> 7 ^ 8 + n + m 2 1 (n m) ! (?i + m) ! 



V ^ M " j ^ : (2-T)T^r-riy(2+ 3) {l.3.5...(2n- 1 )}-' ' 



P /rr\u P /tr\. 7 fi 2n 2 +2n + 27n 2 -2) 

 hence p (H?) 3 dp = (H?) 3 dp \ 1 - -------- 



J-i ./-i I (z,n i)(Zn + 6) } 



n ffff" 1 i r n 1 i 



and ^^^ = |U//;H -5 



.1-1 - W/A J_l J 



(n-m)!(n + m)l . . 



2 " 



Also 



hence (1 - 



(n-m)l(n + m)l \ _ 1_ 3 (tf + n + m?- 1) 



{1.3. 5...(2-l)} 2 \2n+l (2w- 



A. II. 



_ {'ii, m)>. \-ti, i-m; '. n 2 + n 3m 2 / 



Z ri " c /irr-r^p ( 2 n-l) (2n+l) (2 + 3)" 



53 



