386 Lord Rayleigh on the Work done hy Forces 



If, as throughout the present paper, a, /3, 7, &c. be pro- 

 ])ortional to e'^', d^ajdt'^= ^jy^a, and (1) &c. become 



(cr-Z>2>ZS/rf,r + //^V^a+/«-hX' = CK . . (3) 



{a^-h'')dBldi/ -[-h'\/'l3-^p' fi + Y' = 0. . . (4) 



{a^-P)d8/dz{-h^V''y-^2->^y-\-y-^' = (). • • (5) 



These are the fundamental equations. For our purpose 

 we may suppose that X^, Y^ vanish throughout, and that 7j 

 is finite only in the neighbourhood of the origin. It will be 

 convenient to write 



l-=zpfb, h=p/a (6) 



The dilatation B is readily found. Differentiating (3), (4), 

 (5) with respect to .r, y, z and adding, we get 



V2^ + /i23 + a-2,/Z7^L~ = (7) 



The solution of (7) is 



. 1 C^^dTle-^^^ , . . 



%. %y *^ 



T denoting the distance between the element at .r, ?/, z near 

 the origin (0) and the point (P) under consideration. 1£ we 

 integrate partially with respect to z, we find 



the integrated term vanishing in virtue of the condition that 

 7} is finite only within a certain space T. Moreover, since 

 the dimensions of T are supposed to be very small in com- 

 parison with the wave-length, d{r~^ e~''^'')ldz may be removed 

 from under the integral sign. 



It will be convenient also to change the meaning of .r, j/, c, 

 so that they shall represent as usual the coordinates of P 

 relatively to 0. Thus, if Zi e''P* denote the whole force 

 applied at the origin, so that 



Z^^p^S^^Z'd.rdydz, 



in which p is the density, 



5. Zi d fe-'^'-x 



^= ^-rra^pdz^CVh ^^^ 



giving the dilatation at the point P. 



