108 Mr. 0. Heaviside. [June 15, 



The numerical tests in this example are perfectly satisfactory ; and if 

 the numerical meaning of a divergent series could be always as easily 

 fixed, it would considerably facilitate the investigation of the 

 subject. 



Condensed Generalized Notation. Generalization of the Descending 

 Series for the Bessel ^Function through the Generalized Binomial 

 Theorem. 



29. Since the series A and B are particular cases of the general 

 formula (77), Part I, or of 



= 2 W' 



by taking r = and r = \ respectively, it may be desirable to find 

 the general formula of which the series C is a particular case. Notice, 

 in passing, the shorter notation employed in (11). It is certainly easier 

 to see the meaning of a series by inspecting the written-out formula 

 containing several terms, when one is not familiar with the kind of 

 series concerned. As soon, however, as one gets used to the kind of 

 formula, the writing out of several terms becomes first needless, and 

 then tiresome. The short form (11) is then sufficient. One term 

 only is written, with the summation sign before it. The other terms 

 are got by changing r with unit step always, and both ways. The 

 value of r is arbitrary, though of course it should have the same value 

 in every term so far as the fractional part is concerned, so that, in 

 (11), r may be changed to any other number without affecting its 

 truth. Similarly, the exponential formula may be written 



* = *>' (12) 



with r arbitrary and unit step. 



Now, to find the generalized formula wanted, we have, by (25), 

 Part I, 



- 1 )~ l - (13) 



Expand this according to the particular form of the binomial theorem 

 got by taking n = -J in (84), Part I, leaving m arbitrary. Or 

 writing that general formula thus : 



j \r \n-r 



which is compact and intelligible, according to the above explanation, 

 take n = J, and write 2V" 1 in place of x. This makes 



