PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 25 



cosines of 00! ; l\ m, n those of the given force, and put for shortness k for the cosine of 

 the angle between the direction of the force and the line 00, produced, so that 



k = It + mm' + nri . 



Consider at present the first term of the right-hand side of (29). Since the radius vector 

 drawn from O to any element of T ultimately coincides with 00,, we may put I outside the 

 integral signs, and replace da by r~ 2 dS. Moreover, since this term vanishes except when at 

 lies between the greatest and least values of the radius vector drawn from to any element 

 of T, we may replace t outside the integral signs by a~ i r. Conceive a series of spheres, with 

 radii ar, 2ar...nar,... described round 0, and let the n th of these be the first which cuts T. 

 Let S u Sn... be the areas of the surfaces of the spheres, beginning with the » th , which lie 

 within T; then 



ff(9o) at dS - krF(t - nr) S, + krF \t - (n + 1) T } S, 4- ... 



But F(t - nr), F {t - (n + 1) r] ... are ultimately equal to each other, and to 



*(»-3- ° r < Br >-''K> 



and arSi + arS 2 + ... is ultimately equal to T. Hence we get, for the part of f which 



Ik t t\ 



arises from the first of the double integrals, / \t — 1 . The second of the double 



^■n-Da'r \ a) 



integrals is to be treated in exactly the same way. 



To find what the triple integral becomes, let us consider first only the impulse which was 



communicated at the beginning of the time t — nr, where m lies between the limits a^ l r and 



6~'r, and is not so nearly equal to one of these limits that any portion of the space T lies 



beyond the limits of integration. Then we must write nr for t in the coefficient, and 3lq — u 



becomes ultimately {3lk — I') rF(t - nr), and, as well as r, is ultimately constant in the 



triple integration. Hence the triple integral ultimately becomes 



(3 Ik - I') T 



iTrr 3 



nr . rF(t - nr), 



and we have now to perform a summation with reference to different values of n, which in the 

 limit becomes an integration. Putting nr = if, we have ultimately 



t = c^, l.nr.rF(t-nr) = f r ] t' F {t - t') dt'. 



a 



It is easily seen that the terms arising from the triple integral when it has to be extended 

 over a part only of the space T vanish in the limit. Hence we have, collecting all the terms, 

 and expressing F (t) in terms o{f(t), 



> lk j.L r \ l'-lk ,/ r\ Slk-l' r l ,, , , 



To get tj and £, we have only to pass from /, t to m, m' and then to n, n. If we take 

 00 x for the axis of x, and the plane passing through 00i and the direction of the force for 

 Vol. IX. Part I. 4 



