210 Lord Eayleigh on the 



dp= P dY (1) 



If, as we shall here suppose, the matter be arranged in plane 

 strata, the expression for the potential at any point is 



J 00 



(2) 



where p is the density at a distance z from the point in 

 question. Expanding in series, we may write 



so that 



V=2K. p + 2lJ+..., .... (3) 



where 



K = 27rfV(*)<fe, L = 7rf z 2 f(z)dz. ... (4) 

 Jo Jo 



The integrals involving odd powers of z disappear in virtue of 

 the relation ty(—z)=ilr{z). 



We may use (3) to form an expression for the pressure 

 applicable to regions of uniform density (and potential). 

 Thus, integrating (1) from a place where p = pi to one where 

 p = p 2 , we have 



P2-pi=$pdV=[ P Yl-$Yd P 



= 2K(p 2 2 - / 3 1 2 ) - pp {2Kp + 2L d 2 p/dz 2 + ...} 

 = K(p 2 2 -/vO-J dp \2Ld 2 p/dz 2 + ...}• 

 In the latter integral each term vanishes. For example. 



and at the limits all the differential coefficients of p vanish 

 by supposition. Thus, in the application to regions of uniform 

 density — uniform, that is, through a space exceeding the range 

 of the attractive forces, 



Pa—pi=TSHpf—pi*); (5) 



or, as we may also write it, 



p=<GT + Kp 2 , (6) 



where xs is a constant, denoting what the value of p would be 

 in a region where p = 0. We may regard ot as the external 

 pressure operative upon the fluid. Equation (5) may also be 



