4 Prof. J. W. Nicholson on Zonal 



polynomials with factor 1— jjl 2 , are 



i(i_^p m p,Mbgi±j_i(i- : ^)p.p i ';iogi±j 



-xp„P„(l-^)|l 0g i±^. 



The first and second of these three expressions tend to zero 

 at the limits, and thus 



(m 



-n)(ro + n + l)f P n (/*) Q m (fi) dfi = - j~P m P„] 



(_ \m+n__]_^ 



and is zero if m + n is even, and equal to — 2 if m + n is odd, 

 We deduce the two formulae 



I. 

 I 



V(M) P^+iW if. = (2w ,, 2w _ 1)( L + 2» + 2) ' < 4 > 



V + i(/*) *W) «• = ( 2m _ 2 „ + 1)( 2 CT + 2n + 2) ' (5 > 



where m and w are integers. Since when n is odd, Q n is an 

 even function of /m and P w an odd function, and vice versa, 



1 



P»Q,(/,)^ = (6) 



in all cases in which p and ^ are both odd or both even. 



Similar integrals from zero to unity are of importance in 

 the applications mentioned. For these^ we require Q n (0),. 

 Qn'(O). Since 



rx . x t -o i 1 + m- f 2w — 1 -n, 2n — 5 -p. "1 



Q„(/.)=iP»lo gl -^ - ^ T _p,_ 1+ __ I P <M . 4+ . . .j , 



we readily find 



Qo(0)=0, (3,(0)=-!, Q 2 (0)=f, 0,(0) = -ft, 

 corresponding to 



P (0) = 1, P 1 (0) = 0, p 2 (0) = -i P,(0) = 0. 



The values of P,/(0) are coefficients of powers of h in the 

 expansion of A(l + A 2 ) ~ 3 / 2 , the general value being 



p' 2 „(0)=o, PVi(0)=(-)". 3 - 2 5 ;]-;-y i , (8> 



