Harmonics of the Second Type. 11 



Relation to Fourier series. 



The Fourier series representing P„(juO is well known, and 

 can be found at once. For P„(cos 6) is the coefficient of h n 

 in the expansion of 



and expanding each bracket and multiplying directly, we 

 find 



2n * C 1 2n 



P B (cos 6) = 2. 22n{n \ )2 i cos n0+ ^ . -^^cos (n-2) 



1.3 2n.2tt-2 1 



+ 274-2n-1.2n-3 COS ( n " 4 ^ + -.r (26) 



This involves the fact that 



I * P n (cos B) cos (n - 2r) (9 d<9 



2n ! 2rl 2w.2n-2. ... 2w-2r + 2 ■ 



= 7r '2 2 »(n!j 2 '2 2 M>!) 2 ' 2n-l. ... 2n-2r + l * ^ > 



when n — 2r is a positive or negative integer, the integral 

 being zero otherwise w T hen n and r are separately integers. 

 On reduction, with n—2r=m, under these circumstances 



p / n + w + l \ p / n~fflj-l \ 



f * P n (cos 0) cos m<9^0= — ^ 2 / ^ — J- . (28). 



J r / n + m+2 \ r / n-m + 2 \ 



This is the same as the integral 



\ Y H (fi) cos (mcos'V)' ^ o - 



J-i VI7-.A* 



Thus we have 



where , ^ 



A n = — — - I P n (/i) cos (m cos l fi) .--—«■ • \n>m), 



2 J-i Vl-/* 2 



