180 Lord Kelvin on the Reflexion 



p denoting the density of the medium, f , rj, f its displacement 

 from the position of equilibrium (#,y,s), and 8 the dilatation 

 of bulk at (a, y, z) as expressed by the equation 



*-2?J+2 0* 



§ 3. Taking d/dx, d/dy, d/dz of (1), we find 



d' 2 S 

 pjr== (*+*»)?* • (3). 



From this we find 



v -* 8 = Ltfc.(Jp (4 , 



Put now 



e-«H-£v*i • -* + £ V- 2 S; l-6 + £^I (5). 

 These give 



fi + p + f = (6) 



and therefore, eliminating by them £. 9;, £ from (1), we find 

 by aid of (4), 



p d S= n ^> p c w= n ^> p d S= n ^-> w- 



§ 4. By Poisson's theorem in the elementary mathematics 

 of force varying inversely as the square of the distance, 

 we have 



V~ 2 S = -|^; J J J d (volume) • pp ; (8) , 



where S, 8' denote the dilatations at auy two points P and P ; ; 

 d (volume) denotes an infinitesimal element of volume around 

 the point P x ; and PP' denotes the distance between the points 

 P and P'. This theorem gives explicitly and determinately 

 the value of V _2 8 for every point of space when o is known 

 (or has any arbitrarily given value) for every point of space. 

 § 5. If now we put 



f2 =j-v- 2 s; % =|v- 2 a ; &=|v-a ; (9) , 



we see by (5) that the complete solution of (1) is the sum of 

 two solutions, (£ 1? %, fi) satisfying (6) and therefore purely 

 distortional without condensation; and (f 2 , %, f 2 ), ivhich, in 



