PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 13 



with reference to the spheres, to interior. It then expands, and forms the inner boundary of 

 the shell in which the wave of condensation is comprised. It is easy to shew geometrically 

 that each envelope is propagated with a velocity a in a normal direction. 



12. It appears in a similar manner from equations (20) that there is a similar wave, 

 propagated with a velocity b, to which are confined the rotations •&', w", -nr"'. This wave 

 may be called for the sake of distinction, the wave of distortion, because in it the medium 

 is not dilated nor condensed, but only distorted in a manner consistent with the preservation 

 of a constant density. The condition of the stability of the medium requires that the ratio 

 of b to a be not greater than that of ^/3 to 2 *. 



13. If the initial disturbance be such that there is neither dilatation nor velocity of 

 dilatation initially, there will be no wave of dilatation, but only a wave of distortion. If it be 



such that the expressions %d% + r,dy + X^dss and -j-d,v + ——dy + — - dz are initially exact differ- 



UV (JbZ (XZ 



entials, there will be no wave of distortion, but only a wave of dilatation. By making 6 = we 

 pass to the case of an elastic fluid, such as air. By supposing a = eo we pass to the case of an 



incompressible elastic solid. In this case we must have initially $ = and — = ; but in order 



€LZ 



that the results obtained by at once putting a = oo may have the same degree of generality as 

 those which would be obtained by retaining a as a finite quantity, which in the end is supposed 

 to increase indefinitely, we must not suppose the initial disturbance confined to the space T, 

 but only the initial rotations and the initial angular velocities. Consequently, outside . T the 

 expression %dx + rjdy + "^dx must be initially an exact diffei-ential d\|/, where \j/ satisfies the 



equation y\^ = o derived from (14), and the expression — dx + -—dy+-^ dz must be 



CLZ u Z CLZ 



initially an exact differential d^r x , where \^/ l satisfies the equation y \^, = 0. So long as a is finite, 

 it comes to the same thing whether we regard the medium as animated initially by certain 

 velocities given arbitrarily throughout the space T, or as acted on by impulsive accelerating 

 forces capable of producing those velocities ; and the latter mode of conception is equally 

 applicable to the case of an incompressible medium, for which a is infinite, although we cannot 

 in that case conceive the initial velocities as given arbitrarily, but only arbitrarily in so far as 

 is compatible with their satisfying the condition of incompressibility. It is not so easy to see 

 what interpretation is to be given, in the case of an incompressible medium, to the initial dis- 

 placements which are considered in the general case, in so far as these displacements involve 

 dilatation or condensation. As no simplicity worth mentioning is gained by making a at once 

 infinite, this constant will be retained in its present shape, more especially as the results arrived 

 at will thus have greater generality. 



14. The expressions for the disturbance of the medium at the end of the time t are linear 

 functions of the initial displacements and initial velocities; and it appears from (21), and the 

 corresponding equations which determine •&', -^' ', and •ar'", that the part of the disturbance 

 which is due to the initial displacements may be obtained from the part which is due to the 



• See a memoir by Green, On the reflexion and refraction of Light. Camb. Phil. Trans. Vol. vn. p. 2. See also Camb. 

 Phil. Trans. Vol. vm. p. 319. 



