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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 cases 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 

 P as the resultant of the elementary disturbances corresponding to 

 the several elements of P. Let it be required to determine the dis- 

 turbance which corresponds to an elementary portion only of the 

 plane P. In this consists the whole of the dynamical part of the 

 theory of diffraction, if we except the case of diffraction at the com- 

 mon surface of two different media ; the rest is a mere question of 

 integration. Let the time t be divided into equal intervals, each 

 equal to r. The disturbance which is propagated across the plane 

 P during the first interval r occupies a layer of the medium having 

 a thickness vt, if v be the velocity of propagation, and consists of a 

 certain velocity and a certain displacement. By the problem above 

 mentioned, we can find by itself the effect of the disturbance which 

 occupies so much only of this layer as corresponds to a given element 

 dS of P. By doing the same for the 2nd, 3rd, &c. intervals r, and 

 then making the number of such intervals increase and their magni- 

 tude decrease indefinitely, we shall get the effect of the disturbance 

 which is continually transmitted across dS. The result is a little 

 complicated, but is much simplified when certain terms are neg- 

 lected which are only sensible when the radius of the secondary 

 wave is comparable with X, and which are wholly insensible in the 

 physical applications of the problem. The result thus simplified 

 may be enunciated as below : — In the enunciation, the term 

 diffracted ray is used to denote the disturbance in an elementary 

 portion of a secondary wave, diverging in a given direction from the 

 centre ; the plane containing the incident and diffracted rays will 

 be called the plane of diffraction, the supplement of the angle be- 

 tween these two rays the angle of diffraction, and the plane passing 



