ABE EMPLOYED IN THE THEORY OF TERRESTRIAL MAGNETISM. 259 



This is the function represented by P"' m in Gauss's Allgemeine Ilieorie des 

 Erdmagnetismus. (See Gauss's Werke, Band v. p. 142.) 



(n-m)(n-m-l)(n-m-2)(n-m-3) __ 

 2.4.(2n-l)(2n-3) ** 



1.3.5... (2)6-1) rim d m P u 



and -- - 



21. When /u, = l, we have 



<*-/>_ (n + m)S 

 ~ 



_ 

 (n-m)lml'2 m ' 



and in particular 



dP n n(n+l) 



Hence, when //.= !, HI or 



2 



123 n 

 ' ' 



-- - - - --. n 



1.2.3... n _(nl)-2 n 

 " 1.3.5 ~'{2rT-l) ~' (In) \ 



_ 1 . 2 . 3 ... (n-m) d m P n _ (n-m)\n\ 2 >l (n + m)l 

 "~1 . 3 .5 ... 2n-l d* TO ' In \ ' n-m I ml 2 m 



(2n-l) df* TO (In) \ ' (n-m) I 



_ 2 n ~ m (n + in) \n\ _ (n + m) ! 



2?i!m! ~ 2 m m! 1 73 . 5 ... 



And in particular when m = l, 



_ 

 Cr "~ 



2"(?^!) 2 n+j. 1 1.2.3 ... 



~ 



_ 

 (2n)! 2 ' (2n)l 2 " ~2 ' 1 . 3 . 5 ... (2n- I) ' 



22. Again, when p- = 0, first, suppose m = and n = 2r. 

 Then the coefficient of ju," in (/u, 2 -!)" is 



n(n-l)... (r+1). y< 

 1.2.8... r 



332 



