174 PROCEEDINGS OF THE AMERICAN ACADEMY. 



for any given value of t, because the arithmetical mean of the error cor- 

 responding to the function <^ + 80 is »-o + Sxq r= Xq + Xq'. Therefore, 

 (18) becomes 



r 



i 



or, when f is replaced by ^ + x^, 



¥' 



(21 ) ^ ($) + H' ($) =^^ (^ - i') ■ ^> (e) ■ de 



'$' 



for an}' possible value of $. Kow, by assumption IT, (^ (x) is develop- 

 able by Taylor's theorem and, therefore, so is ip (|), that is 







therefore, by (13), (14), and (15), 



f 



i' 







so that 



(22) «^W)=X<-'>'l''l'l^ 



2 



Also, by (13) and (21), 



f+«l 



C+«| 1' 



--fe"-d$'f^{i-i>) -f (t') • j.' + f"-v(i) •si-|"-./'(i) -81. 



i i' 



