218 Methods for the Solution of Special Problems [CH. vm 



Again, if we differentiate equation (160) with respect to /z, 



h (1 - 2V + A 2 )~ 2 = %h n ~ , 

 so that, by combining with (161), 



i 

 Equating coefficients of h n , 



Differentiating (162), we obtain 



o r> 

 Eliminating fju -^ from this and (163), 



(2rc + l)7>=?f^ 1 -^p ..................... (164). 



dyLt OfJU 



By integration of this we obtain 



whilst by the addition of successive equations of the type of (164), we 

 obtain 



_, _ ............... (166). 



254. We have had the general theorem ( 237) 



from which the theorem 



follows as a special case. Or since 



da) = sin 6d6d<f) = dfjLd<t>, 



r l p n ()P m (ridp = ..................... (167). 



J -i 

 f+i 

 To find I I^dj^d/ji, let us square the equation 



o 

 multiply by d/ju, and integrate from /z = 1 to ^ 



