2 Prof. J. W. Nicholson on Zonal 



In the first place, by a usual procedure in the theory of 

 differential equations, we can show at once that 



(m — n) (m + n + l)\ Qm{fJ>) Q»M dfi 



= [(1-/, 2 )(Q )U Q;-Q,Q w /)] 1 _ 1 , 



the accent denoting differentiation with respect to p. We 

 shall suppose throughout that m and n are integers, and 

 that fx lies between the values ±1. 



The right-hand side of this equation becomes, by an easy 

 reduction, 



(1—fi 2 ) X arV n - 2r -i X a s P' M _ 2s _i- (1 — ft 3 ) 2 Or P'»-2r-i 2 a«P„_2»-i 

 i(l_^)P/log^S^P w -2r-i + i(l-^ 2 )P m / logi±^'Sa s Pn-2 S -n 



The functions P and P' are polynomials, and finite for all 

 the values of ji concerned. Thus the first row vanishes by 

 virtue of its factor 1 — p?. The second also vanishes because 

 the limit of 



(W)log^ 

 is zero when fi=±l. We deduce that 



(m — n) (m -+- n + 1) \ Q m Q» dfi = 0, 

 and if m and n are different, 



pQ m Q,^=0. ..... (l) 



* -i 



It is more difficult to discuss the case m = n. The 

 generating function / of the series 



2>Q„ (/*)=/(/*, A), 



which also defines Q n (fi) when n is an integer, is too com- 

 plicated to permit of use after the usual manner adopted for 

 the zonal harmonic P n . Instead, we shall expand Q n {u) in 

 a series of harmonics ot type P n (fi). The possibility of the 

 expansion is evident from the facts that (1) the product 

 Pp(yLt) Pg(ft) is so expansible by Adams's theorem, and 



2) log ' -, except at /jl=+1. must admit such an 



1 — A* 



