PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 15 



by the end of that time a system of displacements represented by rv . By hypothesis, the 

 system was in a position of equilibrium at the commencement of the time - t ; and since the 

 forces are supposed not to depend on the velocities, but only on the positions of the particles, 

 the effective forces during the time r vary from zero to small quantities of the order t, and 

 therefore the velocities generated by the end of the time — t are small quantities of the order 

 t 2 . Hence the velocities — v g communicated at the time destroy the previously existing 

 velocities, except so far as regards small quantities of the order t 2 , which vanish in the limit, 

 and therefore we have nothing to consider but the system of displacements tv . Hence the 

 disturbance produced by a system of initial displacements rv Q is represented by f(t + r) -fit), 

 ultimately ; and therefore the disturbance produced by a system of initial displacements v is 

 represented by the limit of ■r~ 1 {f(t + t) - /(/) } , or by f (t). Hence, to get the disturbance 

 due to the initial displacements from that due to the initial velocities, we have only to differen- 

 tiate with respect to t, and to replace the arbitrary constants or arbitrary functions which 

 express the initial velocities by those which express the corresponding initial displacements. 

 Conversely, to get the disturbance due to the initial velocities from that due to the initial 

 displacements, we have only to change the arbitrary constants or functions, and to integrate 

 with respect to t, making the integral vanish with t if the disturbance is expressed by displace- 

 ments, or correcting it so as to give the initial velocities when t = if the disturbance is ex- 

 pressed by velocities. 



The reader may easily, if he pleases, verify this theorem on some dynamical problem 

 relating to small oscillations. 



15. Let us proceed now to determine the general values of £, n, Y in terms of their 

 initial values, and those of their differential coefficients with respect to t. By the formulae of 

 Section I., £, t], £ are linear functions of $, •&', tu", and ■ar'", and we may therefore first form 

 the part which depends upon $, and afterwards the part which depends upon -Br', ■&", -Br"', and 

 then add the results together. Moreover, it will be unnecessary to retain the part of the 

 expressions which depends upon initial displacements, since this can be supplied in the end 

 by the theorem of the preceding article. 



Omitting then for the present -ar', tst", sr'", as well as the second term in equations (2l), 

 we get from equations (10) and (21), 



*~^su-ifT*™° » 



To understand the nature of the integration indicated in this equation, let be the point 

 of space for which the value of \f/ is sought ; from O draw in an arbitrary direction OP equal 

 to r, and from P draw, also in an arbitrary direction, PQ equal to at. Then F (at) denotes 

 the value of the function F, or the initial rate of dilatation, at the point Q of space, and we 

 have first to perform a double integration referring to all such points as Q, P being fixed, and 

 then a triple integration referring to all such points as P. To facilitate the transformation of 

 the integral (22), conceive PQ produced to Q', let PQ' = s, let d V be an element of volume, 

 and replace the double integral ffF.da by the triple integral h~ l fffF.s' 2 dV, taken between 

 the limits defined by the imparities at<s<at + h, which may be done, provided h be finally 

 made to vanish. We shall thus have two triple integrations to perform, each of which we may 



