1893.] On Operators in Physical Mathematics. Ill 



the other, and could be removed from both sides if it were finite. It 

 must not, however, be removed from (24). "What is asserted is that 

 ox(l-f^)" 1 = 0X0, where the first on the right is (| I)" 1 , and 

 also the on the left. 



Again, if we put r = first in (23), making 



^ _ _!._ _JL_ L 4.__fL_ __i___?H j_ (26) 



I i | - i I . n> * * * i V f 



\n\l |n-l n+I 



and then put n = 1, we get = 0. But if we multiply (26) by \n, 

 making 



we see that the descending series vanishes when n is any negative 

 integer. That is, it is asserted that 



(l + oj) = l + na+ ^~W- . . . , (28) 



unless w is negatively integral. But when it is a negative integer 

 there are additional terms, though always in indeterminate form ; for 

 instance, oo XO when n = 1 and x is finite. It would appear, how- 

 ever, that the value is zero, because there is every reason to think 

 (28) correct (as a particular form) in the limit. 



On the other hand, if we multiply (26) by | 1, and so make it 

 cancel the | 1, |-2, &c., in the denominators, we get, when n is 1, 



(l+x)~ l = 1 x + x>- +0- 1 0?-*+ or 3 , (29) 



which is quite inadmissible, since the right member is the sum of 

 two series previously found to be equivalent to one another, and to 

 the left member. The right member is therefore twice as great as 

 the left. 



Improved Statement of the Binomial Theorem ivith Integral Negative 



Index. 



' 



32. A consideration of the above obscurities suggests the following 

 way of avoiding them. We should recognize that the zeros (l^)" 1 

 and r, when we take n = I and r = 0, are independent, and may 

 have any ratio we please. Thus, first put n = 1 + 5 in (23), 

 making: 



