210 Dr. L. N. G. Filon on a New Mode of 



can be expanded in a convergent series of powers of 



for ~ ., I can be kept < 1 at all points of the contour. 



We then find 



But in the case of the figure of eight there must be points 

 on the contour nearer to t=fjb than the two branch-points. 

 This is obvious, since the contour must cut the line joining 

 the points ivA''^ — 1 ^^^^^y^^^ these points. At these inter- 

 sections 1 ^y ij?—i I > I t^fx |. 



But further, the loop must also cut the same line outside 

 the points, and at these intersections | ^fju^ — l \ < | t^fju \ . 



Hence the same expansion will not serve for all parts of 

 the contour, and the method fails. 



19. Some of the associated Legendre's functions of the 

 second kind, however, can be obtained by expanding the 

 function 



1 



t — fjb- 

 For if we write this 



-^<^^-D 



^=- 



^l-,j:\l-y?)+u(t-iJL) 



then, if u and t be small enough, -^ can be expanded in an 

 absolutely convergent series of positive powers of u and t. 



And in virtue of the differential equation (23) satisfied by 

 •^, the coefficient of u'^V'^ will be in all cases a solution of the 

 differential equation of the associated Legendre functions. 



The denominator of -^ may be broken up, as before, into 

 factors, and we find 



, _ —'2u^/l—^x? ^^^^.^^^^^ 



