PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 11 



Section II. 



PROPAGATION OF AN ARBITRARY DISTURBANCE IN AN ELASTIC MEDIUM. 



9. The equations of motion of a homogeneous uncrystallized elastic medium, such as 

 an elastic solid, in which the disturbance is supposed to be very small, are well known. 

 They contain two distinct arbitrary constants, which cannot be united in one without adopting 

 some particular physical hypothesis. These equations may be obtained by supposing the 

 medium to consist of ultimate molecules, but they by no means require the adoption of such 

 a hypothesis, for the same equations are arrived at by regarding the medium as continuous. 



Let x, y, z be the co-ordinates of any particle of the medium in its natural state ; £, r),t 

 the displacements of the same particle at the end of the time t, measured in the directions of 

 the three axes respectively. Then the first of the equations may be put under the form 



dt 2 \dx< dy* dz 2 ) K ' dx \dx dy dz)' 



where a 2 , b 2 , denote the two arbitrary constants. Put for shortness 



dp dn dt k 

 dx dy dz 



and as before represent by v£ tne quantity multiplied by b 2 . According to this notation, the 

 three equations of motion are 



or ay 



(18). 



It is to be observed that 8 denotes the dilatation of volume of the element situated at the 

 point (x, y, z). In the limiting case in which the medium is regarded as absolutely incom- 

 pressible 8 vanishes; but in order that equations (18) may preserve their generality, we must 

 suppose a at the same time to become infinite, and replace a 2 8 by a new function of the co- 

 ordinates. If we take — p to denote this function, we must replace the last terms in these 



dp dp dp 

 equations by — — , — — , — -— , respectively, and we shall thus have a fourth unknown 

 dx dy dz J 



function, as well as a fourth equation, namely that obtained by replacing the second member of 



(17) by zero. But the retention of equations (18) in their present more general form does not 



exclude the supposition of incompressibility, since we may suppose a to become infinite in the 



end just as well as at first. 



10. Suppose the medium to extend infinitely in all directions, and conceive a portion of it 

 occupying the finite space T to receive any arbitrary small disturbance, and then to be left to 

 itself, the whole of the medium outside the space T being initially at rest ; and let it be required 

 to determine the subsequent motion. 



2 — 2 



