10 Prof. J. W. Nicholson on Zonal 



Thus 



i 1 PsnQi) , /1. 3....2n --l\ 8 f 1 _dfM_ 



J-zVW 1 / \ 2. 4.. .. "2n "j J^vr^' 

 leading at once to the result. Moreover, 



J_i\'l-V^ \6.D....2n + l/ J^s'l-^r K } 

 The last integral is readily evaluated. For 



1. LL 



and therefore 



r-^^=r(icos^io g f±^-i)^ 



J_! ^1-,J? ^ Jo \ 2 ^1-COS^ / 



= 1 COS01ogCOt-^# — 7T 



Jo 



= 2 I cos 2^ (log cos 0— log sin 0) d0—ir. 

 Jo 

 Now when r is odd, it is known from a familiar result in 

 the integral calculus that 



C cos2r0logcot0d0=^. 

 So that, in the present, case r = l, we find 



J^l-/* 2 ^ 2 



It follows that for all integer values of n, 



J. 



1 Qj^L ^ = Q, ..... (24) 



.iVr-v 



a somewhat surprising result, which is a special case of a 

 formula in the next section. It can be expressed also in the 

 form p tt 



Q n (cos0)d0 = O, (25) 



.'ft 



Jo 



