THE THEORY OF TERRESTRIAL MAGNETISM. 419 



1 

 Hence 



(2rT-"l) (2n+ 1) (2n+~3) (% -m)I ' 



4 n' + n-3m 2 (n + m}\ 



/n-v 7 

 ~ /AWn) ^ ; ~- 



~ 



J -i ( ~ 



Similarly 



5) 



in all other cases | fj - ^ (?'; Q dp. or | p.'Q^Q^dp. vanishes. 

 J-i \ / J -i 



n 



The value of uf^U^dp, may be derived from the expression for 

 J-i 



ri 



V?QQl\diJ. by multiplying by the ratio 



HH, _ (n-m)\(n l -m)\ 



~ 



_ 

 fg; == 1 . 3 . 5 ~(2w- 1) 1.3.5... (2n,-l) ' 



( l ( l 2\/rr\ 2 7 4 - - 



hence J^ ^-p,j(H n ) dp- -- ( -^ ( ^n+ 1)^+3) {l . 3 . 5 ... (2n- I)} 2 ' 





11. From Art. 1 above (p. 402) we have 



532 



