562 Mr. stokes, ON THE CRITICAL VALUES OF 



happens. For this purpose it is only requisite to take for f(n, h) - U„, {U„ being the sum of the 

 first n terms of (66),) a quantity which has different limiting values according to the order in which 

 w and h are supposed to assume their limiting values, and which has for its finite difference a 

 quantity which vanishes when n becomes infinite, whether A be a positive quantity sufficiently small 

 or be actually zero. 

 For example, let 



f(n,h)-U„^^^, (68), 



nh + I 



which vanishes when re = 0. Then 



A \nn,h) - U„] = v„,. - M„„ = („^ + ,)(;,,+A + l) • 

 Assume U„=l , so that u„ = A?7„_, = 



n(n + 1) 



and we get the series 



+ ... + —, r+ ..., (69). 



1.2 2.3 n(n + 1) 



(70). 



1 + 5h A(A + 2)w^ + A(4-A)w + l-A 



2(1 + h) '" n(«+ 1) {(« - 1) A + 1} (n/t + 1) 



which are both convergent, and of which the general terms become the same when h vanishes. 

 Yet the sum of the first is 1, whereas the sum of the second is 3. 



If the numerator of the fraction on the right-hand side of (68) had been pwA instead of 

 2«/i, the sum of the series (70) would have been p + 1, and therefore the limit to which the sum 

 approaches when h vanishes would have been p + 1. Hence we can form as many series as we 

 please like (67) having different quantities for the limits of their sums when h vanishes, and yet 

 all having their n"* terms becoming equal to ti„ when h vanishes. This is equally true whether the 

 series (66) be convergent or divergent, the series like (67) of course being always supposed to be 

 convergent for all positive values of h however small. 



39. It is important for the purposes of the present paper to have a ready mode of ascertaining 

 in what cases we may replace the limit of (67) by (66). Now it follows from the following theorem 

 that this substitution may at once be made in an extensive class of cases. 



Theorem. The limit of V can never difi"er from U unless the convergency of the series (67) 

 become infinitely slow when h vanishes. 



The convergency of the series is here said to become infinitely slow when, if n be the number 

 of terms which must be taken in order to render the sum of the neglected terms numerically less 

 than a given quantity e which may be as small as we please, n increases beyond all limit as A 

 decreases beyond all limit. 



Demonstration. If the convergency do not become infinitely slow, it will be possible to find 

 a number w, so great that for the value of A we begin with and for all inferior values greater than 

 zero the sum of the neglected terms shall be numerically less than e. Now the limit of the sum of 

 the first M| terms of (67) when A vanishes is the sum of the first w, terms of (66). Hence if e be the 

 numerical value of the sum of the terms after the w,"" of the series {66), U and the limit of V cannot 

 differ by a quantity so great as e + e. But e and e may be made smaller than any assignable 

 quantities, and therefore U is equal to the limit of V. 



Cob. 1. If the series (66) is essentially convergent, and if, either from the very beginning, or 

 after a certain term whose rank does not depend upon A, the terms of (67) are numerically less than 

 the corresponding terras of (66), the limit of V is equal to U. 



