388 REPORTS ON THE STATE OP SCIENCE. — 1919. 



then 



<f>{a,)=fic,),cf,{-a,)=f{c:), 



and we finally have 



^'f{x)dx^Up-q) |^Ao/(co) + S A, {f{c,) + f{cj)} ' . 



(li) Let n be even and equal to 2q, where q is an integer. The roots 

 of P„(?/) are y = ± b,. (for r=l, . . . , g) ; so P„(?7) is equal to 



Thus 



P„'(&,) =2Mb,n{b,^-b,% 



and 



The value of constant A, corresponding to the root y — b^oi P„(^)=0, is 



r {y+bAAM:zK)dy, 

 Ji 2b,u{b:^-b,'^) 



which, for the same kind of reason as before, is equal to 



J-i n{b;'-bP) 



that is, 



, ^ 1 r 1 ib^ ib^^ \ 



' n(V-&,^)l2?-l 2g-3 + 22-5 ••7- 



The value of the constant A, corresponding to the root y= —b^ of 

 P^ (^)=0, is (on evaluation) the same as that of the constant A^. Thus, 

 as before, but with these values of the constant A„ we have 



|V(x)dx=i(p-g)[2 A. I / (c ) + /(O I ], 

 with the same literal significance for c, and c/, viz., 



It should be noted that, in this expression, 



2A.=1. 

 3. Some of the simplest cases will now be taken. 

 I. Let 71=8, so that^=l. The equation P.,(2/)=0 gives 



5y,-3tj=0 

 ao that ai2=f . Then 



^ / y 2\ .^ 8 



A ^ X S . 



•'*-l — — 2 * 3 — ^ ' 



