On Taylor's Theorem. 191 



greater than <$a; and if 4/(0 + 0//) be always negative, f (« + //) 

 is less than <pa. 



Theorem 8. If $a = and tya = 0, and if ^ (0 + 6//) be 



always positive, then tt^^ must be equal to some value 



r$\a±jh) 

 V{a + Qh)' 



Theorem 9. If <px and tyx be two functions of x which 

 satisfy the following conditions : 



$ a = $' a = <p n a = . $<"J « = 



$£ = 4/'« = \[/"« =x \[/<"> a = ; 



and if 



<| (a + Qh) t[/(a + M) *<» +1 >(a + M) 



be severally always positive when A is positive ; then 



H«+a) .u i. i f ^ (w+i) («+^) 



-71 — — it must be equal to some value of \. J , 1 ,\ -^L 



The preceding conditions are satisfied by 



<J> x = fx —fa — (x—a) fa - x — a J — 

 fa 



szJ** , x f {n) a 



fyx = (* — a) n+1 , 

 and we then have 



<P(a + h) /%H' (#+•♦*) 



4»(A+i) : 2.3... (n+iy 



or 



/(« + A) =fa+fa.h + .... +/<«>« 



2 



the only condition being that fa, fa, ...y^ a must not be 

 either of them infinite. 



This is the form in which Lagrange exhibited the theorem, 

 and it never Jails, as it is customary to say when a theorem 

 which has been exhibited in too general a form shows sym- 

 ptoms of discontent at being asked to perform all that its god- 

 fathers promised for it. 



I believe that any one may establish the preceding series of 



