E.vpressino Solutions of Laplace's Equation. 199 



is then also convergent, if | /x | < 1, which is the case 



in physical applications, where //, = cos 6. The convergence of 



the series for (l- y^^Jf )"^ requires ^'^^-^^' < 1, or 



^^ Since the greatest value of //,+ Vl — yu,^ is 



\/2 when yu- is the cosine of a real angle all the conditions for 

 the validity of the expansions are satisfied if t < — =. 



In like manner, when t is large, we can use an expansion 

 in inverse powers of t, and we find 



= ^Tl^lprp - befove. . (18) 



9. AVe can, however, generalize this result and obtain all 

 the associated Legendre coefficients as the coefficients in the 

 expansion of a function of two variables. 



Remembering the result 



,kA-u)= 2 u'^^„^{z) . . . (19) 



(see Gray and Mathew's ' Bessel Functions,' p. 17), 



let us write in this z= ^y/T-^^ and apply this operator 



1 ^^ 



to the tunction , 



t — jjb 



Yil-^ — '-Xj 



Now apply the symbohc form of Taylor's theorem to the 



left-hand side and expand , — —on the rioht-hand side in 

 ^ t—p. ® 



powers of 1//, t being supposed large. 



= 2 2 ^;^^^P;r(;.),by(16). . (21) 



111= —X n = () ^ ^ 



Hence the associated Leoendre's coefficient P"' is oiven as a 



