ON gauss's theorem for quadrature, etc. 387 



The value of A, corresponding to the root y=a, of P„(?/)=0 is 



But the integral 



is zero, because the subject of integration is an uneven function of y 

 consequently 



J-i2a/n(a/-a,2) 

 a/n(a/-a,^) ^^i^ + l 2^-1 2^-3 •• ' /' 



S 



where, in the sums, 2 denotes that a/' must not occur. 



The value of A corresponding to the root y= — a of P (7/)=0 is 



J-i 2a,ll(a/-a;^) 



Because of the vanishing integral 



r\ 



yn{y'-a;')dy, 



-1 



the value of this coefficient A is 





that is, it is equal to A, 

 Thus \?e have 



I <A(2/)%=Ao</.(0)-hiA, (<^ (a,) + (/,(-«,) I, 

 J-i s=i L J 



with the stated values of Ao, A„ . . . , A.,, in the case when n is equal 

 to the uneven integer 2^9 + 1. 

 It should be noted that 



Ao + 22A, = 2; 



an immediate inference from the simplest form of <h(y) taken as a 

 constant. 



Reverting to the earlier integral with the ic-range from p to a we 

 write 



Co Up + q), 



