170 Compressible and N on- Homogeneous Masses [CH. vn 



This value of E may be regarded as being obtained by a double inte- 

 gration with respect to <f and X. In fig. 35, let OA, OB represent axes 



of q and X respectively, and let the thick 

 curve PQ represent the relation between q 

 and X expressed by equation (442). This 

 curve meets the axis of q at the value q = q', 

 for by the definition of q', we have 



* +^ 



where X = 0. 

 at X = oo . 



It clearly meets q = (p = p ) 



Thus it appears that the first integral in 

 the value of E is represented by an inte- 

 gration over the area BQR while the second 







Q A 



Fig. 35. 



integral is represented by integration over the area RQAS. Thus the whole 

 integration is over the area which is shaded in the figure, and on changing 

 the order of integration we find 



where the lower limit q is now determined as a function of X by equation 

 (442). 



This completes the evaluation of F t -. The external potential can be 

 evaluated in a similar way*, but is not required in the present problem. 



Configurations of Equilibrium 



171. We may now turn to the conditions of equilibrium, which as we 

 have seen ( 166) are expressed by the single equation (424), namely 



C , . 



p 1 - 



.(448). 



In this equation p has the value 



- ...... (4*9). 



Expanding py~ l by the binomial theorem, we find that equation (448) 

 assumes the form 



* See Bakerian Lecture for 1917. Royal Society Phil. Trans. A (not yet published). 



