274 Prof. J. R. Young on the development of a Binomial. 

 of which the factors involving x, when «/3 is put for x, are 



n{n—m) m ~ n -^ m - n i and (n — m) m - n fi m - n ; 



so that the expression for the wth differential coefficient, still 

 omitting the numerical factor 1.2.3 ... «, is 



(m — l)(m — < 2)....(m — n + l) , . . . 



A i.g.jfrj.!) — {-(n-™) m ~ n +{n-™) m - n }P m - n > 



which, when n < m, is zero. If n = m, the first term of the 

 general coefficient of h n above, is simply unit : hence, introdu- 

 cing the absent numerical factor, we have in this case 



(A.)=1.2.3...w, 



as was to be proved. 



The expression under the sign of differentiation in (A.), if 

 developed, and then the series differentiated n times, w/3 being 

 afterwards put for x, will furnish new properties of the binomial 

 coefficients, which need not however be here formally exhibited. 



There are probably other expressions, analogous to (A.), 

 that might suggest developments of like generality with that 

 of Abel; but I question whether they are worth searching 

 for. In fact, and the remark is of some importance, all such 

 general theorems must be received with caution and quali- 

 fication : they are strictly true only when the terms of the series 

 become at length actually zero, or at least tend to zero. With- 

 out this qualification, the binomial theorem itself, even in its 

 ordinary form, would be fallacious. Analysis, unfortunately, 

 abounds with fallacies of this kind. Theorems are sometimes 

 presented to us with an aspect of unlimited generality, which 

 are not true even in a single individual case ! The following 

 is an instance : viz. 



<p(x) = <p(x-\-*) +<?{%— «)— <p(# + 2a) — <p{x — 2a) + .... 

 the simplest case of which, viz. 



x=x+x— X— X + X + .... 



is repugnant to common sense. 



This theorem was, I believe, first given by Francais in the 

 Annales des Math., vol. iii. It was again obtained by Abel, 

 in vol. ii. of his (Euvres, p. 85, and has been reproduced by 

 Gregory in his Examples, p. 243. It is the tendency of ana- 

 lysis, if left unrestrained, to spread, in some directions, into 

 unprofitable luxuriance, by which it is encumbered, but not 

 advanced. A little in the way of correcting this fault, I have 

 myself attempted in the pages of this Journal and elsewhere; 

 but much more is required. This, however, is a task which 

 I leave to those whose position may be lesc inimical to the 



