A.S COMMONLY INVESTIGATED. 241 



Now, repeatedly substituting x + h for x, we obtain 



F (x + n h) = F (x) + n ^ (h). F x + R 

 and, on the contrary, by putting n h for h in the expansion of F (x + h,) we 

 also deduce 



F (x + n h) = F (x) + ^n h) F X + R. 



Whence the right hand members of these two equations may be equated; 



and as each is arranged in the order of its magnitude, we must have, by what 



has preceded, the first terms of the same magnitude, and consequently only 



differing by functions of magnitudes higher than their own. And from this, 



again, it follows that, by removing from the first to the higher terms in either 



series, magnitudes that shall leave the order undisturbed, we may render the 



first terms in the two series identical. Whence, writing k for n h, making h 



^h 

 constant, and denoting i_ by c; and finally, representing, still, by ^k what £k 



becomes after removing the terms in question, we have 



ck = ^k. 



And as the constant c may be taken from ^ k, and made a factor of F x, it 

 appears, at length, that we have 



{F(x + h) = Fx + h. F. x + R.} 



X = X . . . X 

 h = s . . . £ 



where s and e are quantities as small as may be necessary. 



Denoting F x, according to the usual notation, by d F x, or, more simply, by 



s 



d F., the equation 



I F (X + h) = F. + dj. + R J (9) 



manifestly suffices to establish all that is commonly taught in our text books 

 concerning first differentials, as well as so much of the theory of the higher 

 differentials as regards them merely as the differentials of differentials, and not 

 as the superior terms in Taylor's Theorem. 



The nature of these last might now be investigated by a method very similar 

 to that which Poisson employs for the same purpose, but the process may be 

 vii. — 3 L 



