386 REPORTS ON THE STATE OF SCIENCE. — 1919. 



The integral becomes 



h(P-l) I ^(y)dy. 



The equation giving the positions of the selected ordinates is 



that is, the positions are given by the roots of 



where P„ is the Legendre function (the Zonal Harmonic) of order n. 

 All the n roots are known to be real, and they lie between +1 and —1. 

 If n is even, the roots are equal and opposite in pairs. If n is odd, one 

 root is zero and the remaining roots are equal and opposite in pairs. 



2. Let tti, . . . , a„ be the roots of P„ (y) =0 ; then 



P„(^)=Nn(2/-a,), 



where the constant N is immaterial to our purpose. The Lagrange 

 formula of interpolation now is 



and so 

 where 



J-1 y — o-r P,'(a,) 

 Two cases arise, according as n is odd or is even. 



(i) Let n be odd and equal to ^p + l, where p is an integer. The 

 roots of P„(2/)=0 are y=0, and y=±a, (for r=l, . . . , p) ; so 



P„(2/) = %(2/— a,2)(2/2-a,2) . . . (y'-a/). 

 Thus 



P,/(0) = N(-l)"a/^a,:^ . . . a/, 



P„'(a,) = 2Na,2ri(a;^-a;^), 



where 11 denotes the product of a/ — a,'^, a,- — 02", . . 

 with the omission of the factor a/- — a/- ; and 



«»"— V' 



P,/(-a.)=2Na.2n(ar'-a,2). 



The value of Aq, corresponding to the root y=0 of P„(z/)=0, is 



J-i a, . . . Up 



SO that 



A =9 (-1)" / J_ _ W , 2a^a./ _ 1 



' "a, 2 . . . a/\2p+l 2p-l "^ 2p-3 • * ■ J 



