124 Mr. 0. Heaviside. [June 15, 



the genera] results. We also come to confirm the idea we have 

 recognized that equivalence may be understood in three distinct senses. 

 viz., analytical, algebraical, and numerical. The first use made. by me 

 of equivalent series, one of which is continuously divergent, was 

 analytical only. The second use was numerical. The third is alge- 

 braical, through the generalized algebraical theorems. We also see 

 that equivalence does not necessarily or usually mean identity, 

 Thus the series A, B, C are analytically, algebraically, and numeri- 

 cally equivalent with x positive. Bat they are not algebraically 

 identical. The identity is given by C = (A + B). This point is 

 rather important in some transformations, and explains some pre- 

 viously inexplicable peculiarities. Thus, the series A is real whether 

 x be real or a pure imaginary. In the latter case, we get the oscil- 

 lating function J (#), the original Fourier cylinder function. But 

 the equivalent series C becomes complex by the same transformation. 

 The above-mentioned identity explains it. The* second solution of 

 the oscillating kind is brought in, as will appear a little later ( 70). 



Partial Failure of Interpretation of Numerical Value of Divergent Series 

 by Initial Convergence. Further Explanation yet required. 



48. (g). But whilst we thus greatly extend our views concerning 

 divergent series, the question of numerical equivalence, which just 

 now in (/) seemed to be about settled, becomes again obscured. The 

 property that the value of a divergent series, including the con- 

 tinuously divergent, may be estimated by the initially convergent 

 part, is a very valuable one. But the property is not generally true, 

 and, in fact, sometimes fails in a very marked manner. We must, 

 therefore, reserve for the present the question of numerical equiva- 

 lence in general, and let the explanation evolve itself in course of 

 time. If definitely understandable numerical equivalence of series 

 were imperative under all circumstances, then I am afraid that the 

 study of the subject would be of doubtful value. But the matter has 

 not this limited range, a very important application of divergent 

 series being their analytical use, which is free from the numerical 

 difficulty. For example, the extreme forms of the binomial theorem 

 may, when considered numerically equivalent, be utterly useless. Yet 

 they may be employed to lead to other series, either convergent, or it 

 may be divergent, but with a satisfactory initial convergence con- 

 trasting with the original. Note that the series may sometimes take 

 the form of definite integrals, apparently of infinite or of indefinite 

 value. In any case we should not be misled by apparent unintel- 

 ligibility to ignore the subject. That is not the way to get on. We 

 have seen the error fallen into by Boole and others on the subject of 

 divergent series. It is not so long ago, either, since mathematicians 



