12 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



Differentiating equations (18) with respect to w, y, «, respectively, and adding, we get 

 by virtue of (17) 



%-^- <•»■ 



Again, differentiating the third of equations (18) with respect to y, and the second with respect 

 to z, and subtracting the latter of the two resulting equations from the former, and treating in 

 a similar manner the first and third, and then the second and first of equations (18), we get 



~SF'^^ -JT^-"' ^~ = bvw ' ■ • (20) - 



where ■&', w", ■&"' are the quantities defined by equations (13). These quantities express the 

 rotations of the element of the medium situated at the point {x, y, z) about axes parallel to the 

 three co-ordinate axes respectively. 



Now the formula (7) enables us to express $, w', ■&", and w'" in terms of their initial values 

 and those of their differential coefficients with respect to t, which are supposed known ; and 

 these functions being known, we shall determine f, y, and £ as in Art. 7- Our equations 

 being thus completely integrated, nothing will remain but to simplify and discuss the formulae 

 obtained. 



11. Let be the point of space at which it is required to determine the disturbance, r the 



radius vector of any element drawn from O ; and let the initial values of $, — be represented 



at 



by f(r), F(r), respectively, with the same understanding as in Art 4. By the formula (7), 



we have 



*-$- RF (?*)*«+ ±-%-tfff{at)d* . . . .(21). 



47T 47T at 



The double integrals in this expression vanish except when a spherical surface described 

 round O as centre, with a radius equal to at, cuts a portion of the space T. Hence, if O 

 be situated outside the space T, and if r ir r 2 be respectively the least and greatest values of 

 the radius vector of any element of that space, there will be no dilatation at until at = »y 

 The dilatation will then commence, will last during an interval of time equal to a -1 (r 2 - r,), 

 and will then cease for ever. The dilatation here spoken of is understood to be either positive 

 or negative, a negative dilatation being the same thing as a condensation. 



Hence a wave of dilatation will be propagated in all directions from the originally dis- 

 turbed space T, with a velocity a. To find the portion of space occupied by the wave, we 

 have evidently only got to conceive a spherical surface, of radius at, described about each 

 point of the space T as centre. The space occupied by the assemblage of these surfaces is 

 that in which the wave of dilatation is comprised. To find the limits of the wave, we need 

 evidently only attend to those spheres which have their centres situated in the surface of the 

 space T. When t is small, this system of spheres will have an exterior envelope of two 

 sheets, the outer of these sheets being exterior, and the inner interior to the shell formed by 

 the assemblage of the spheres. The outer sheet forms the outer limit to the portion of the 

 medium in which the dilatation is different from zero. As t increases, the inner sheet con- 

 tracts, and at last its opposite sides cross, and it changes its character from being exterior, 



