200 Theory of a Non-Conservative Gas [CH. ix 



239. Let fig. 16 represent a plane in which all values of t', real and 

 imaginary, are represented, and let A OB be the axis along which t' is real, 

 being the point t' = 0. 



A o B 



FIG. 16. 



If P represents the point 



t' = Re ie 



R (cos + i sin 0), 



then the angle FOB will be equal to 6, and OP will be equal to R. 

 By Cauchy's Theorem of contour integration, 



(485), 



where the integral is taken round any closed contour in the plane of fig. 16, 

 and ^Z is the sum of the residues of f(t') inside this contour. Let us take 

 f(t') = Ue 1pt> , and take the closed contour to consist of a semicircle BPA of 

 radius R described about the origin as centre, and the real axis AOB which 

 forms the bounding diameter of the semicircle. In the limit we shall take 

 =00. 



We have supposed as a legitimate first approximation, that U vanishes 

 except between t' = and t' = T. But when we attempt to represent the 

 conditions of nature by strict mathematical analysis, there can be no dis- 

 continuity of this kind and the function U' must be supposed to exist from 

 t = oo to = oo . This latter supposition, it is true, includes the former as 

 a special case, but the special case is not only excluded on physical grounds, 

 but is inconsistent with the supposition made in equation (485) that U is an 

 analytic function. When in the limit we make R = oo , the real axis A OB 

 will extend from oo to + oo so that the integral of Ue ipt> taken along this 

 real axis may be supposed equal to the integral taken through the encounter, 

 and therefore, by equation (480), to ap(X + iY). 



The remainder of the left-hand member of equation (485), the integral 

 being taken round the prescribed contour, is the integral taken round the 



