110 Mr. 0. Heaviside. [June 15, 



Employing this in (18), we can reduce the series to one containing 

 integral powers only. The coefficient of x is made to be 



That this reduces correctly to a convergent series summing up to ^, 

 when r = J, may be anticipated and verified. Also, that when 

 r = we obtain unity is sufficiently evident. In these conclusions 

 we merely corroborate the preceding. But I have not been able to 

 reduce (22) to a\ simple formula showing plainly in what ratio the 

 formulae A and B are involved when r has any other values than 

 and -| (or, any integral value, and the same plus -J). 



The Extreme Forms of the Binomial Theorem. Obscurities. 



31. There are some peculiarities about the extreme forms of the 

 binomial theorem when the exponent is negative unity (or a negative 

 integer) which deserve to be noticed, because they are concerned in 

 failures, or apparent failures, which occur in derived formulae. These 

 peculiarities are connected with the vanishing of the inverse factorial 

 for any negative integral value of the argument. Thus, in 



(! + #)" ^ x r 



-\r- = s j7f^7' (23) 



take n = I. We obtain 



(24) 



Now, on the left side we have the vanishing factor (| I)" 1 . So, on 

 the right side, the quantity in the big brackets should generally 

 vanish. This asserts that 



1 _ aj + a .2_ a 53 + = a,-i_aj- 3 +aj- 3 - 4 + , (25) 



where on the left side we have the result of dividing 1 by 1 + x, and, 

 on the right, the result of dividing 1 by oj + 1, or x~ l by 1-f x~ l . 

 These series are the extreme forms of the expansion of (l + x)~ l by 

 the ordinary binomial theorem, and they are asserted to be algebraic- 

 ally equivalent, although the numerical equivalence, which is some- 

 times recognisable, is often scarcely imaginable. 



But observe that if we choose r = as well, we have a nullifying 

 factor on the right side also of (24). It is apparently the same as 



