1893.] On Operators in Physical Mathematics. 123 



analytical operators to obtain convergent algebraical solutions. The 

 series C and C' above referred to are then truly the two independent 

 solutions of a certain differential equation, and neither should be 

 rejected, for they are natural companions. 



Disappearance of the Distinction 'between Direct and Alternating 

 Divergent Series. 



46. (e). But, still pursuing the subject along the same lines, this 

 view is soon found to be imperfect. For a given operator leading to 

 a convergent solution one way may lead to a divergent solution by 

 another. Or it may lead to the same algebraical function by diverse 

 ways. These and other considerations show that divergent series, 

 even when continuously divergent, must be considered numerically 

 as well as algebraically and analytically. But in the analytical use 

 of a direct or continuously divergent series every term must be used, 

 if the result is a convergent series. Yet it is plain that we cannot count 

 the whole divergent series numerically, because it has no limit. And 

 on examination we find that the initial convergent part of the con- 

 tinuously divergent series gives the value of the function in the same 

 sense, as an alternatingly divergent series. In the latter case we 

 come nearest to the value by stopping at the smallest term, where 

 the oscillation is least. If we now make all terms positive, so that 

 the series is continuously divergent, and treat it in the same way, 

 and stop when the addition made by a fresh term is the smallest, 

 we come near the true value. 



We now seem to have something like a distinct theory of divergent 

 series. The supposed distinction between the alternating and the 

 continuous divergent series has disappeared. Analytical equivalence 

 of two series, one convergent, the other divergent, may require all 

 terms in the divergent one to be counted. Numerical equivalence 

 exists also, but is governed by the initial convergency. 



Broader and Deeper Views obtained by the Generalized Calculus. 

 Analytical, Numerical, and Algebraical Equivalences. Equivalence 

 not necessarily Identity. 



47. (/). The last view is a distinct advance, and it is certainly 

 true in the case of many equivalences, including some which are of 

 importance in mathematical physics. But, again, further examina- 

 tion shows that the last word has not been said. For on seeking to 

 explain the meaning and origin of equivalent series, we are led to a 

 theory of generalized differentiation, involving the inverse factorial 

 as a completely continuous function both ways, and to methods of 

 multiplying equivalent forms to any extent, and in a generalized 

 manner, all previous examples being merely special extreme cases of 



