Expressing Solutions of Laplace's Equation. 201 



Substitute now in the difFerential equation the expand 



•ion 



^ -^ (72 + m)! ^"^^ , 



and equate to zero the coefficient o£ ?:'"r ''-^ in tiie result ; we 

 obtain the differential equation satisfied by the associated 

 Leoendre's coefficient, namely, 



11. The last result enables us to obtain various expressions 

 for P- 



For the expression 



2{t-co^e) / i\ 



may be broken up into factors 



rher( 



a2-l 



a. 



— ? 



) 2^— cos_6') ^ 1 

 ) ""sin 6 J 



2 2a(^— COS 6) _ 



f-cos^ ±^t'^~2t con 0+1 

 sin 6 



Let us take that sign of the square root which makes 



I a I =co when \t \ =go. If then ^f — 2^ cos ^+1 denote 



that branch of the function which is real and positive when t 



is large, real, and positi^-e we have to take the + sign and 



we have 



^^ ^ -Gos (9 -f- y/^^- 2t co s e~+ 1 

 sm6' 

 Then 



1 2a, cosec 



(25) 



■co^ 



^-^^'("-«)"(^-")(-]) 



(26) 



