or Centiifugal Theory of Elasticity, 359 



radius of curvature of the path of the particles through {x, y, z) ; 

 then the above equations obviously become 



p ax ax r 



(3A) 



P ay dy r 



p dz az r 



If these equations are integrable, 



— dx+ —dy+ ^dz 



must be an exact differential. Let — be its primitive function, 

 the negative sign being used, because a!, /3', 7' must be generally- 

 negative. Then the integral of the equations (3) is 



log^ p = j^J — = -r (2Q0 — ^)+ constant ; 



or taking p, to denote the pressure at the bounding surface of 

 the atom, 



20 1 



-^(?»-(f,)-i(*-*,) 



P = Pie ' (4) 



Our present object is to determine the superficial-atomic den- 

 sity p„ and thence the pressure p = hp^, in terms of the mean 



density ^f and heat Q. For this purpose we must introduce the 



above value of p into equation (2)^ giving 



f^=p\ JJJ ^ f <^^ ^y ^^> 



whence 



2Q 



p = hp^== f^l^^jJJ^ f dx dy dz. (5 ) 



Let the volume of the atom be conceived to be divided into 

 layers, in each of which has a constant value. Then we may 

 make the following transformations : — 



ffJdxdydz=Kmfe'''-'' t.dcf> 1 



jyye-^^''-^^^dxdydz=kMYfe''''-''^,d<f>] ' ^^^ 



k being a specific constant, and yjr and co functions of <f), and of 

 the nature and density of the substance. 



