112 Mr. 0. Heaviside. [June 15, 



-l+s-r / _2 + ,-r 



1 --- * - - 



H oTMH- ^fy -^"Mi-l- 1. (30) 



s r \ (s r)(l + sr) \ j 



This being general, let r and s be both infinitely small, but without 

 any connexion. We know that the rate of increase of the inverse 

 factorial with n is 1 when n is 1. It follows that 



These, used in (30), make it become 



, 1 + s r 



_ r \ " j ' (32) 



Ultimately, therefore, we obtain in a clear manner 



*--.... \(3S| 



This seems to be the proper limiting form of the binomial theorem 

 when the index is negative unity. It asserts that the two extreme 

 equivalent forms may be combined in any ratio we please, since r/s 

 may have any value. If r = 0, we have the ascending series only. 

 If r = 5, then the descending series only. If s = 2r, we obtain half 

 their sum. The expansion is indeterminate, but the degree of indeter- 

 minateness appears to be merely conditioned by the size of the 

 ratio r/s. 



We may also notice that the suppositions that 5 is infinitely small 

 and r is finite, so that 



used in (30), lead us to 

 (1+ a)- 1 = - l-a>+a J z -a? 8 + .. . . 



(34) 



that is, the difference of the two extreme equivalent series divided 

 by 0, which is, of course, indeterminate. 



