250-253] Spherical Harmonics 217 



so that on picking out coefficients of h n , 



D 1.3...2n-l - , - 1.3...2n-3 



Pn= 2.4...2n 2cQS ^ + ^ 2 . 4 ... 2n _ 2 2cos("-2)0+.... 



Every coefficient is positive, so that 7^ is numerically greatest when each 

 cosine is equal to unity, i.e. when = 0. Thus P n is never greater than 

 unity. 



Fig. 75 shews the graphs of 7?, 1$, 7J, 7J, from //.= 1 to /*= + !, the 

 value of being taken as abscissa. 



Relations between coefficients of different orders. 

 253. We have 



Differentiating with regard to h, 



(160). 

 (161), 



so that 



-h)(l + 2h n P n ) = (1 - 2V + /i 



Equating coefficients of A 71 , we obtain 



........... (162). 



This is the difference equation satisfied by three successive coefficients. 



