14 Prof. J. W. Nicholson on Zonal 



, . u m (n — m + 2) (n + m — 1) 



and since — — = —. -^—, ~ , 



u m-2 (w — m+l)(n + ?n) 



we have Um _ 2 = = u n - S . 



Thus ( '"q,, (cos 0) cos (?i-l-2r) 0.dd = O . (35) 



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for integer values of n and r. 



Now let 7? be even, and written as 2n. If m is also even, 

 and written 2 in, 



f* 



^2™= 1 Q2» (cos 0) cos 2?n0 <i# 

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 _ _ (2tt — 2m + 2)(2n + 2 m-l) 

 ~ U2m - 2 (2n-2m-hl)(2n + 2m) ' 



■ and so down to u , which is zero as might be foreseen, Q 2n 

 being odd, and cos 2m6 even in //,. 



Thus f* Q 2n (cos 6) cos 2m6 dd = Q 



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for all integer values of m and n. Similarly, 



1 * Q2n+i (cos 6) cos (2m + 1) 6 dd = 0. 



The equation (35) has a remarkable consequence. For 

 since 



£* Q n (eos0)cos(n-l-2r)0d0 = O, 



-and also, the integrand being an odd function, 

 f * Q n (cos ffj cos (n - 2r) rf0 = 0, 



f 7 " Q ;l (cos 0) cos m<9 6$ = . . . (36) 

 for aZZ cases in which m<n. A special case is, of course, 



f'Q n (cos^^= P -%^L^=0, 



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already proved, whether 7i be odd or even. 



In obtaining Fourier series for the Q functions, our 



it follows that 



