32 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



Now produce 0' O l to 2 , and refer O x a>, x y, O x %, O x 2 , O x O to a sphere described 

 round O x with radius unity. Then x0 2 forms a spherical triangle, right angled at 2 , and 



x0 2 = w, 2 0= — + 0, 0% = <p, 0%0 2 => — + ■%, whence we get from spherical trigo- 

 nometry, 



cos d> = — sin 6 sin w, sin ^> cos ^ = cos 6, sin <p sin ^ = cos 6 tan ^ = sin 6 cos to. 



We have therefore 



£ = s sin cos sin a>, >? = — s sin 2 9 sin a> cos a>, £ = s (l — sin 2 sin 2 a>). 



To find the aggregate disturbance at O, we must put for s its value, and perform the 

 double integrations, the limits of m being and 2ir, and those of r being \Zp* and eo . The 

 positive and negative parts of the integrals which give £ and t) will evidently destroy each 

 other, and we need therefore only consider £. Putting for s its value, and expressing in 

 terms of r, we get 



£= — //(r +p) (r 2 cos 2 u> + p 2 sin 2 w) cos— - (bt - r) — - — . . (47) 



Let us first conceive the integration performed over a large area A surrounding 0', which 

 we may afterwards suppose to increase indefinitely. Perform the integration with respect to r 

 first, put for shortness F (r) for the coefficient of the cosine under the integral signs, and 

 let R, a function of w, be the superior limit of r. We get by integration by parts 



fF (r) cos — (bt - r) dr - - — F (r) sin — (bt - r) + (—) * F' (r) cos— (bt - r) + ... 

 X 27r X \2 ir I X 



Now the terms after the first must be neglected for consistency's sake, because the formula 

 (46) is not exact, but only approximate, the approximation depending on the neglect of terms 

 which are of the order X compared with those retained. The first term, taken between limits* 

 gives 



~F(±p)si a ^(btTp)-~F(R)sin^(bt-R), 

 2w X 27T X 



where the upper or lower sign has to be taken according as lies in front of the plane 

 P or behind it. We thus get from (47) 



t--0 ±l)sin— (bt*?p) - — f ' F(R) sin— (bt-R) da,. 



2 X 47T^0 X 



When R becomes infinite, F (R) reduces itself to cos 2 w, and the last term in £ becomes 



4 t J 



2tt 

 .ecs 2 u> sin — (bt — R) dw. 



Suppose that no finite portion of the perimeter of A is a circular arc with O' for 

 centre, and let this perimeter be conceived to expand indefinitely, remaining similar to itself. 

 Then, for any finite interval, however small, in the integration with respect to m, the function 



sin — (bt - R) will change sign an infinite number of times, having a mean value which is 

 X 



