AS COMMONLY INVESTIGATED. 243 



change their order, and by annexing these parts to terms higher in the series, 

 we might at length render 



h.d^F = dH.P; 



and by cancelling these equal terms, and reasoning in the same manner, arrive 

 ultimately at the equation 



n — In — 1 an 



H d P = d . H P. 



X h 



The first of these two equations is satisfied by the functions 



I U2 1 



H = iL, p=.d^F. 



s 



and the second by 



n — 1 n — 1 n n — 1 



H = dH,P = d.P. 



h X 



whence we conclude that 



n — 1 n — 1 



H = j-J—^ h", P = d-F. 

 and that one of the prime developments of F (x + h) is 



F + h.d F + ^d^F + &c 



X 



A very common, and, it appears to me, a universal error committed in demon- 



h" 

 strating this theorem, consists in regarding the function ^-^ — - as the only 



classifying function by which such an expansion could be made; whereas not 



z 



only is 2 A h^* a function that equally satisfies our differential equations, 



z = n I 



but, as we have already remarked, those very equations are unessential to the 

 problem, since any function of the order h° could have been made to supply its 

 place. This error, it appears to me, lies not so much in a want of knowledge 

 of the fact here stated, as in a total omission of that fact in the demonstrations 

 of the theorem, whereby it necessarily follows that such demonstrations are 

 made to prove too much. 



In concluding this part of my subject, I may, perhaps, be permitted to detain 

 the attention of the reader for a moment, by referring him to an error somewhat 

 similar in the usual extensions of Newton's Binomial Theorem. This ex- 

 pansion, when confined to integral and positive indices, expresses merely 

 that special case of the equivalence 



n — 1 



n n— 1 1 



* Ah" + A h "~^ + &c Ah* + A. 



