concept of II inform convergence 155 



' 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 series numerically less 

 than a given quantity e, which may be as small as we please, n 

 increases beyond all limit as h decreases beyond all limit. 



'Demonstration. If the convergency do not become in- 

 finitely slow it will be possible to find a number n, so great that 

 for the value of h tue begin with and for all inferior values greater 

 than zero the sum of the neglected terms shall be numerically less 

 than e....' 



Stokes's words, and in particular those which I have italicised, 

 seem to me to make two things perfectly clear. 



(1) Stokes is considering neither a property of an interval 

 (a, b) im Grossen (such as is contemplated in A 1 or B 1), nor a 

 property of a single point which (as in A 3 or B 3) need not be 

 shared by any neighbouring point, but a property of an interval 

 im Kleinen, that is to say a small but fixed interval chosen to in- 

 clude a particular point. His definition is therefore one of the 

 type of A 2 or B 2. 



Stokes's failure to perceive the bearing of his discovery on 

 problems of integration is made much more natural when we 

 realise that he is considering throughout a neighbourhood of a 

 point and not an interval im Grossen. And this remark applies 

 to Seidel as well. 



(2) Stokes is considering an inequality satisfied for a special 

 value of n, or at most an infinite sequence of values of oi, and not 

 necessarily for all values of n from a certain point onwards. In 

 this respect there is a quite sharp distinction between Stokes's 

 work and Seidel's. What Stokes defines is (to use the language 

 of this note) a mode of quasi-unifo7'ni convergence and not one of 

 strictly uniform convergence. 



It seems to me, then, that what Stokes defines is what I have 

 called quasi-uniform convergence in the neighbourhood of a, point 

 (B2). 



8. If we adopt this view, Stokes's mistake becomes very much 

 more intelligible. He proves, quite correctly, that uniform con- 

 vergence in his sense implies continuit}^ : his proof, stated quite 

 formally and by means of inequalities, is substantially that given 

 in 1 5, under (1). He then continues* as follows. 



' Conversely, if (66) is convergent, and if U= Vof, the con- 

 vergency of the series (67) cannot become infinitely slow when h 



* p. 282. Tbe italics are mine. 



t Ffl is what Stokes calls 'the value of V for h = 0', by which he means, of 

 course, its limit when h tends to 0. 



