;,:>,; Mr. 0. Heaviside. [Feb. 2, 



I make to be 2815'71 by the ordinary formula, and 2815-75 by (78), 

 including the semi-convergent part. But the last two figures are 

 probably wrong, as there is a good deal of figuring involved in the 

 calculation of both (72) and the convergent series in (78). 



When smaller values of x are taken, the numerical agreement per- 

 sists as far as the initial convergency of the descending series permits, 

 as in the case of the series (31), for example. Later on I will co- 

 ordinate (78) and (72) with the descending formula (31). 



The companion formula to (78) is 



2/1 1 . 1 2 3 2 1 2 3 2 5 2 . \ 2 / C 3 . X* 



We might expect this to be a form of the oscillatory function J (a?). 

 But it is not. It represents the oscillatory companion to J (#), say 

 GO (a), which maybe exhibited in an ascending series of the whole 

 powers of x z together with a logarithm, so standardized as to vanish 

 at infinity. This function will appear later. The double series (79) 

 occurs in Lord Rayleigh's ' Sound.'* The series (78) I have not come 

 across. 



The Binomial Theorem Generalized. 



23. Let us next generalize the binomial theorem in a similar 

 manner. We have 



a* 



"" 



in ascending powers of x. Or 



But in descending powers of a, which is the only other form gener- 

 ally known, we have 



(g + l) = g a*" 1 g-* 



J j \n l J2|n 2 



- (82) 



That we may use the generalized exponential we might infer from 

 forms (80) and (82) being equivalent, combined with previous 



be*readily dedlTd 



