PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 3 



The supposition of in compressibility reduces the two arbitrary constants to one; but as the 

 equations are not thus rendered more manageable, I have retained them in their more general 

 shape. 



The first problem relating to an elastic medium of which the object that I had in view 

 required the solution was, to determine the disturbance at any time, and at any point of an 

 elastic medium, produced by a given initial disturbance which was confined to a finite portion 

 of the medium. This problem was solved long ago by Poisson, in a memoir contained in the 

 tenth volume of the memoirs of the Academy of Sciences. Poisson indeed employed equations 

 of motion with but one arbitrary constant, which are what the general equations of motion 

 become when a certain numerial relation is assumed to exist between the two constants which 

 they involve. This relation was the consequence of a particular physical supposition which he 

 adopted, but which has since been shewn to be untenable, inasmuch as it leads to results which 

 are contradicted by experiment. Nevertheless nothing in Poisson's method depends for its 

 success on the particular numerical relation assumed ; and in fact, to save the constant writing 

 of a radical, Poisson introduced a second constant, which made his equations identical with the 

 general equations, so long as the particular relation supposed to exist between the two constants 

 was not employed. I might accordingly have at once assumed Poisson's results. I have how- 

 ever begun at the beginning, and given a totally different solution of the problem, which will 

 I hope be found somewhat simpler and more direct than Poisson's. The solution of this 

 problem and the discussion of the result occupy the first two sections of the paper. 



Having had occasion to solve the problem in all its generality, I have in one or two in- 

 stances entered into details which have no immediate relation to light. I have also occasionally 

 considered some points relating to the theory of light which have no immediate bearing on 

 diffraction. It would occupy too much room to enumerate these points here, which will be 

 found in their proper place. I will merely mention one very general theorem at which I have 

 arrived by considering the physical interpretation of a certain step of analysis, though, properly 

 speaking, this theorem is a digression from the main object of the paper. The theorem may 

 be enunciated as follows. 



If any material system 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, the part 

 of the subsequent motion which depends on the initial displacements may be obtained from 

 the part which depends on the initial velocities by replacing the arbitrary functions, or 

 arbitrary constants, which express the initial velocities by those which express the correspond- 

 ing initial displacements, and differentiating with respect to the time. 



Particular cases of this general theorem occur so frequently in researches of this kind, 

 that I think it not improbable that the theorem may be somewhere given in all its generality. 

 I have not however met with a statement of it except in particular cases, and even then the 

 subject was mentioned merely as a casual result of analysis. 



In the third section of tbis paper, the problem solved in the second section is applied to 

 the determination of the law of disturbance in a secondary wave of light. This determination 

 forms the whole of the dynamical part of the theory of diffraction, at least when we confine 

 ourselves to diffraction in vacuum, or, more generally, within a homogeneous singly refracting 



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