830 Mr. J. Herapatli on the Binomial Theorem, 



finity with unequal terms : we may also, from the develop- 

 ments of 



fd^{x,y)=f[d^, + dy ]^ {x,y) (14) 



f and 4) implyhig any functions whatever and d^ d diffe- 

 rentials respectively relative to x and y, deduce some valu- 

 able consequences; but such speculations, to do them justice, 

 would much surpass the reasonable limits of a mere sketch 

 like this. I shall therefore now merely develop rf"'' 7/^, and 



proceed to general differences. Suppose in (14) fd = rf~% 

 and (p{x,y) =■ x° u then by our demonstration of the bino- 

 mial theorem 



d~''u = 0_/u^-r.Oxdu + r~ .Ox'"^^d^u ... (15) 



_r— 1 ■' — r— 2 X 



whatever be the value of r, supposing dx= I. Ifr=l the 

 expression coincides with Bernouille's. This is the natural 

 integral ; if the complete be sought it is found by merely adding 

 the correction of (12) or (13), as ?• may be an integer or fi-ac- 

 tion to (15). This is the complete integral of any function 

 of X whatever be its order found in the successive differentials 

 of the given function. It however depends on the form of 

 u whether this expression be very useful. Other and much 



more serviceable formulae might be given, by considering the 

 function as of two variables; but as they would lead me into 

 discussions foreign to my present object, which chiefly re- 

 spects non-integral differentiation, I shall not swell the paper 

 with them. 



On General Differences. 

 When the order of the difference is a positive integer, the 

 formula (4) is probably as good as any that could well be 

 given ; but if the order be fractional or negative it is better to 

 have recourse to (6) or (7). For instance, n being negative 

 we have 



4- 2 -I- 9, ... (16) 



when t — \. But in this case it is obvious that all the mem- 

 bers become infinite. Now I have already observed in nega- 

 tive differentials, that the integrals obtained by our process 

 are uncorrected. The same is also true in negative diffe- 

 rences. If we th -""efore partially correct the preceding ex- 

 pression by differentiating each term with respect to t in the 



* I here employ 2,^, 3^^ . . to signify the binomial coefficients. 



numerator 



