180 Compressible and Non- Homogeneous Masses [CH. vn 



We may notice that although terms in 7 occur in the equation to the 

 boundary, yet no such terms occur in equations (487) and (488). This is 

 necessarily the case, as can be at once seen on considering the incompressible 

 mass for which 7= oo . For such a mass, we have already seen ( 171) that 

 c (po <r) and therefore also 7 (p g <r) remain finite, while the value of 

 &> 2 /27T/5 must necessarily reduce to '18712. 



Equation (488) indicates that compressibility increases the value of &> 2 /27rj5 

 which is necessary for the pseudo-spheroidal form to give place to a pseudo- 

 ellipsoidal form. Stated in another way, a compressible mass retains the 

 form of a figure of revolution up to higher values of the rotation than does 

 an incompressible mass, rotation being measured with reference to the mean 

 density p. 



180. We have so far obtained a solution which is accurate as far only as 

 the first order of the small quantity p Q cr. The method we have used 

 admits of extension as far as any power of p - &, but the labour of com- 

 putation makes it almost impossible to carry the calculations beyond second 

 order terms. 



To obtain a second order solution, we may replace eP by eP + e 2 Q, and 

 Aw 2 by Aft) 2 + Sft> 2 , where So> 2 is of the second order. Similarly we replace 

 AFi by AF; + SFi. We are accordingly assuming a boundary of the form 

 (cf. 172) 



l ........................ (489), 



corresponding to a rotation ft> given by 



0,2 = ^2 + Aw 2 + 3ft) 2 ........................ (490). 



The general equation (461), written down as far as second order terms, 

 now becomes 



On equating the first order terms in this equation we of course obtain 

 equation (462). On equating the second order terms we obtain 



