EXPANSION 0¥ F {X + h). 161 



With regard to the negative values of ^, we shall have, by theorem (2), 

 ^-^F' {a — h) > 0, F' {a — h) — By 0, 

 Ah'-F{a — h) + Fa<0, F{a — h) + B h — Fa <, 0, 

 or F{a — h)yFa—Ah,F{a — h)<,Fa — Bh; 



and the reasoning continued as before will lead to a similar result. 



It may be observed, in conclusion, that the integrations effected above are perfectly allowable', 

 since the equation 



F{x-^ h) = Fx + h.F'x + h.R 

 is sufficient for all purposes of differentiation and integration. And it is immaterial whether any 

 other primitives than those obtained exist, since we are not seeking the only expansion of F{x -f h),- 

 but one true expansion of it. (See Calc. des Fonctions, Le^on 9me.) 



VIII. — 2 Q 



