118 Mr. 0. Heaviside. [June 15, 



giving in the case of x = 1, 



1'815 1 1'880 



u = 1'880, v = - : . ' . I = - - = 1'035. 

 Jo ' \J5_ 1>815 



Thus we have practically gone over the ground from n = J to = 1 

 with good results, so far as the limited examples are concerned, and 

 there can be, so far, scarcely a doubt of the existence of numerical equi- 

 valence, in the same sense as before with respect to the ascending and 

 descending series forthe Fourier-Bessel f unction. It remains to examine 

 cases between n = -J and 1. This is important on account of the 

 complete failure in the latter case of the numerical equivalence when 

 estimated in the above manner. From the already shown indeter- 

 minateness of the binomial expansion when n = 1, we have the 

 suggestion of a partial explanation, because we should arrive at the 

 form au + bv, where a + b = 1. But there remains the fact in- 

 dicated that the extreme, forms of the binomial expansion are 

 equivalent, so that we should expect u and v to be equivalent. Since, 

 however, the numerical equivalence of the different forms of (l + x) H 

 becomes very unsatisfactory when n is or is near 1, so we should 

 not be surprised to find that the unsatisfactoriness becomes empha- 

 sized in the case of u and v. Such is, in fact, the case. 



Failure of Numerical Equivalence of Derived Series reckoned by Initial 

 Convergence, at first slight, and later complete, when n approaches a 

 Negative Integer. 



39. Take n = -f in (37), (38). Then 



When x = 1, we find that 



0-25 or 0-39 

 u = 0-497, v = -- - - , 



according as we do not, or do, count the smallest term in v. That is, 



1 0-497 



|| " 0'25 or 0'39 * 



Now the first gives far too great a result, whilst the other, though not 

 so bad, is still too great. That is, the v series gives too small a 

 result, when the smallest term is fully included. A part of the next 

 term is needed, to come to u. 



