168 



PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



In this paper I have given three examples of the method just described. The first relates 

 to the integral W, the second to an infinite series which occurs in a great many physical inves- 

 tigations, the third to the integral which occurs in the case of diffraction with a circular 

 aperture in front of a lens. The first example is a good deal the most difficult. Should the 

 reader wish to see an application of the method without involving himself in the difficulties of 

 the first example, he is requested to turn to the second and third examples. 



FIRST EXAMPLE. 

 1. Let it be required to calculate the integral 



cos — (w 3 — mw) dw 



2 v J 



0) 



for different values of m, especially for large values, whether positive or negative, and in par- 

 ticular to calculate the roots of the equation W= 0. 



2. Consider the integral 



U= [ -(cos30 + V-lsin36)e* s -»#) d x 



(2) 



where 9 is supposed to lie between - - and + — , in order that the integral may be con- 

 vergent. 

 Putting 



as = (cos 9 - \/ — l sin 6) z, 



we get dx = (cos0 - V— 1 sinfl) dz, and the limits of % are and » ; whence, writing for 

 shortness 



p = (cos 2 9 + <s/ - 1 sin 2 9) n, (3) 



we get 



u 



= (cos 9 - v/^Tsin 9) f e-0*-P*) dz*. 



•'A 



(4) 



3. Let now 9, which hitherto has been supposed less than — , become equal to -. The 



6 6 



integral obtained from (2) by putting 9 - — under the integral sign may readily be proved to 



* The legitimacy of this transformation rests on the theo- 

 rem that if /(*) be a continuous function of x, which does not 

 become infinite for any real or imaginary, but finite, value of x, 

 we shall obtain the same result for the integral of f(x) dx be- 

 tween two given real or imaginary limits through whatever 

 series of real or imaginary values we make x pass from the 

 inferior to the superior limit. It is unnecessary here to enun- 

 ciate the theorem which applies to the case in which f(x) be- 

 comes infinite for one or more real or imaginary values of x. 



In the present case the limits of x are and real infinity, and 

 accordingly we may first integrate with respect to z from to a 

 large real quantity «, , 6' (which is supposed to be written for 6 

 in the expression for x) being constant, then leave x equal to z„ 

 make 6' vary, and integrate from 6 to 0, and lastly make #, infi- 

 nite. But it may be proved without difficulty, (and the proof 

 may be put in a formal shape as in Art. 8,) that the second inte- 

 gral vanishes when ar, becomes infinite, and consequently we 

 have only to integrate with respect to z from to real infinity. 



