PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 29 



several elements into which we may conceive the plane P divided. Let it then be required to 

 determine the disturbance corresponding to an elementary portion only of this plane. 



In practical cases of diffraction at an aperture, the breadth of the aperture is frequently 

 sensible, though small, compared with the radius of the incident waves. But in determining 

 the law of disturbance in a secondary wave we have nothing to do with an aperture ; and in 

 order that we should be at liberty to regard the incident waves as plane all that is necessary is, 

 that the radius of the incident wave should be very large compared with the wave's length, 

 a condition always fulfilled in experiment. 



32. Let 0, be any point in the plane P; and refer the medium to rectangular axes 

 passing through 0„ x being measured in the direction of propagation of the incident light, 

 and * in the direction of vibration. Let/ (bt - x) denote the displacement of the medium at 

 any point behind the plane P, x of course being negative. Let the time t be divided into 

 small intervals, each equal to t, and consider separately the effect of the disturbance which 

 is transmitted across the plane P during each separate interval. The disturbance transmitted 

 during the interval t which begins at the end of the time t' occupies a film of the medium, 

 of thickness br, and consists of a displacement/^/) and a velocity bf (bt'). By the formulae 

 of Section II. we may find the effect, over the whole medium, of the disturbance which exists 

 in so much only of the film as corresponds to an element dS of P adjacent to X . By doing 

 the same for each interval t, and then making the number of such intervals increase and the 

 magnitude of each decrease indefinitely, we shall ultimately obtain the effect of the disturbance 

 which is continually propagated across the element dS. 



Let O be the point of the medium at which the disturbance is required ; I, m, n the 

 direction-cosines of X measured from 0„ and therefore — /, — m, — n those of 00 x measured 

 from O; and let 00^ = r. Consider first the disturbance due to the velocity of the film. 

 The displacements which express this disturbance are given without approximation by (2.Q) 

 and the two other equations which may be written down from symmetry. The first terms 

 in these equations relate to normal vibrations, and on that account alone might be omitted 

 in considering the diffraction of light. But, besides this, it is to be observed that t in the 

 coefficient of these terms is to be replaced by a~ x r. Now there seems little doubt, as has 

 been already remarked in the introduction, that in the case of the luminiferous ether a is 

 incomparably greater than b, if not absolutely infinite*; so that the terms in question are 

 insensible, if not absolutely evanescent. The third terms are insensible, except at a distance 

 from O] comparable with X, as has been already observed, and they may therefore be 

 omitted if we suppose r very large compared with the length of a wave. Hence it will be 

 sufficient to consider the second terms only. In the coefficient of these terms we must replace 

 t by b~ x r ; we must put w = 0, v Q = 0, w = bf (bt - r), write - I, — m, - n for I, m, n, and 

 put q = —nw = — nbf (bt — r). The integral signs are to be omitted, since we want to 

 get the disturbance which corresponds to an elementary portion only of the plane P. 



" I have explained at full my views on this subject in a paper On the constitution of the luminiferous ether printed in 

 the 32nd volume of the Philosophical Magazine, p. 349. 



