106 PROFESSOR STOKES, ON THE DISCONTINUITY 



arbitrary constants in the ascending and descending series. In the examples to which I have 

 applied the method, with one exception, this was effected, so far as was necessary for the 

 physical problem, by means of a definite integral, which either was what presented itself in 

 the first instance, or was employed as one form of the integral of the differential equation, and 

 in either case formed a link of connexion between the ascending and the descending series. 

 The exception occurs in the case of Mr Airy's integral for m negative. I succeeded in 

 determining the arbitrary constants in the divergent series for m positive ; but though I was 

 able to obtain the correct result for m negative, I had to profess myself (p. 177) unable 

 to give a satisfactory demonstration of it. 



But though the arbitrary constants which occur as coefficients of the divergent series may 

 be completely determined for real values of the variable, or even for imaginary values with 

 their amplitudes lying between restricted limits, something yet remains to be done in order to 

 render the expression by means of divergent series analytically perfect. I have already 

 remarked in the former paper (p. 176) that inasmuch as the descending series contain radicals 

 which do not appear in the ascending series, we may see, a priori, that the arbitrary con- 

 stants must be discontinuous. But it is not enough to know that they must be discontinuous ; 

 we must also know where the discontinuity takes place, and to what the constants change. 

 Then, and not till then, will the expressions by descending series be complete, inasmuch as 

 we shall be able to use them for all values of the amplitude of the variable. 



I have lately resumed this subject, and I have now succeeded in ascertaining the character 

 by which the liability to discontinuity in these arbitrary constants may be ascertained. I may 

 mention at once that it consists in this ; that an associated divergent series comes to have 

 all its terms regularly positive. The expression becomes thereby to a certain extent illusory ; 

 and thus it is that analysis gets over the apparent paradox of furnishing a discontinuous 

 expression for a continuous function. It will be found that the expressions by divergent 

 series will thus acquire all the requisite generality, and that though applied without any 

 restriction as to the amplitude of the variable they will contain only as many unknown con- 

 stants as correspond to the degree of the differential equation. The determination, among 

 other things, of the constants in the development of Mr Airy's integral will thus be rendered 

 complete. 



1. Before proceeding to more difficult examples, it will be well to consider a com- 

 paratively simple function, which has been already much discussed. As my object in treating 

 this function is to facilitate the comprehension of methods applicable to functions of much 

 greater complexity, I shall not take the shortest course, but that which seems best adapted to 

 serve as an introduction to what is to follow. 



Consider the integral 



u = 2 f 



Jo 



e'"' sin2ax(Lv = H (l) 



1 2.3 3.4.5 ' 



