112 PROFESSOR STOKES, ON THE DISCONTINUITY 



the sign being + or - according as the amplitude of a lies within the limits and - , or 



— and — . It is worthy of remark that in this expression the transcendental quantity tt^ 

 appears as a true radical, admitting of the double sign. 



Two cases of the integral j e'''da occur in actual investigations, namely when = — , 



when the integral leads to / e~*'dt, which occurs in the theory of probabilities, and when 



•'o 



0=—, when it leads to Fresnel's integrals T' cos (-«') ds and / sin (-«'] ds. In the 



latter case the expression (11) is equivalent to the development of these integrals which has 

 been given by M, Cauchy. 



11. If in equation (11) we put a = p (cos ± v - 1 sin 9), where is a small positive 

 quantity, and after equating the real parts of both sides of the equation make 9 vanish, we 

 find, whichever sign be taken, 



1 1.31.3.5 , rf 



- + h + 



= 2e-'''r^'dp (13) 



p 2/3' ay 2'p' 



The expression which appears on the second side of this equation may be regarded as a 

 singular value of the sum of the series 



1 1 1.31.3.5 



a "^ 2^' ■*" 2^ "•■ ~2W" "^ ^^*^ 



a series which when 9 vanishes takes the form of the first member of the equation. The 

 equivalent of the series for general values of the variable is given, not by (13), but by (11). 

 It may be remarked that the singular value is the mean of the general values for two infinitely 

 small values of 9, one positive and the other negative. 



These results, to which we are led by analysis, may be compared with the known theory 

 of periodic series. If /"(a?) be a finite function of a?, the value of which changes abruptly 

 from a to 6 as a? increases through the value c, a quantity lying between and ir, and /{x) 

 be expanded between the limits and tt in a series of sines of multiples of a?, and if (p {n, x) 

 be the sum of n terms of the series, the value of (p (n, x) for an infinitely large value of n and 

 a value of a> infinitely near to c is indeterminate, like that of the fraction 



(tv + yy + x - y 

 {x -yy + 0! + y' 



which takes the form - when a? and y vanish, but of which the limiting value is wholly 



indeterminate if a? and y are independent. We may enquire, if we please, what is the limit 

 of the fraction when <t? first vanishes and then y, or the limit when y first vanishes and then x, 

 for each of these has a perfectly clear and determinate signification. In the former case 

 we have, calling the fraction >|/ (tv, y). 



