236 Cambridge Philosophical Society. 



of diffraction, which will be mentioned further on, the illumination 

 in front of an aperture is insensible except in the immediate neigh- 

 bourhood of a normal to the primary wave drawn through a point 

 in the aperture. Consequently we are only concerned with the law 

 of disturbance in that part of a secondary wave which lies very near 

 the normal to the primary wave ; the nature of the disturbance in 

 other directions does not affect the result, since the secondary waves 

 neutralize each other by interference. Now it has been shown by 

 others, by indirect methods, that if c be the coefficient of vibration 

 in the incident light, d^ an element of the area of the aperture, r the 

 radius of a secondary wave diverging from f?S, A the wave length, 



the coefficient of vibration in the secondary wave will be - — , and 

 the phase of vibration must be accelerated by a quarter of an undula- 

 tion ; or in other words, - must be subtracted from the retardation 

 4 



due to the radius r. These results, however, according to what has 

 been already remarked, only apply to that portion of a secondary 

 wave which lies immediately about the normal to the primary. The • 

 object of the author in this paper was to determine, on purely dyna- 

 mical principles, the law of disturbance in any direction in a se- 

 condary wave. 



The author has treated the aether as an elastic solid ; and as such 

 it must be treated in considering light, if the theory of transverse 

 vibrations be not rejected. The object which he had in view re- 

 quired the solution, in the first instance, of the following problem ; — 

 to determine the disturbance at any point of an elastic medium, and 

 at any time, due to a given small arbitrary disturbance confined to 

 a finite portion of the medium. This problem was solved long ago 

 by Poisson ; but the author has given a totally different solution of 

 it, which appears to be in some respects simpler than Poisson's. In 

 the course of the solution, the author was led incidentally to the fol- 

 lowing very general dynamical theorem. 



Let any material system whatsoever, in which the forces acting 

 depend only on the positions of the particles, be slightly disturbed 

 from a position of equilibrium, and then left to itself : then the part 

 of the disturbance at any time which depends on the initial displace- 

 ments will be got from that which depends on the initial velocities 

 by differentiating with respect to the time, and replacing the arbi- 

 trary functions, or arbitrary constants, which express the initial ve- 

 locities by those which express the corresponding initial displace- 

 ments. Particular caBes of this theorem are of frequent occurrence, 

 but the author is not aware of any writing in which the theorem is 

 enunciated in all its generality. 



The problem above-mentioned has been applied by the author to 

 the case of diffraction in the following manner. Conceive a series 

 of plane-waves of plane-polarized light propagated in a direction 

 perpendicular to a fixed mathematical plane P. According to the 

 principle of the superposition of small motions, we have a perfect 

 right to consider the disturbance of the aether in front of the plane 



