1883.] On the Figure of Equilibrium of a Planet, fyc. 161 



Integrating this, with the condition that the pressure vanishes at 

 the surface, we have, 



= 2: [i 



whence the modulus of compressibility is 



lV-^-=^7T W 1 n . 



dw n(\. — Tfii) 



3 



The case of n=l is interesting; it gives a constant modulus of 

 compressibility equal to 27r, and the law of pressure p = 27r log w. 



If n be less than unity the compressibility, or reciprocal of the 

 modulus, increases with the density, which is of course physically 

 improbable. If n be greater than unity and less than 3, the com- 

 pressibility becomes less the greater the density. The assumed law 

 probably does not give such good results as those of Laplace, because 

 the decrease of compressibility with increasing density is not 

 sufficiently rapid. 



The range of n=S to n=l gives the results which possess most 

 physical interest. 



In comparing results with those of Laplace there will be occasion 

 to express the modulus as a length ; that is to say, we are to find the 

 length of a column of unit section whose weight (referred to the 

 surface gravity of the planet) is equal to the force specified in the 

 modulus. 



Now if g be gravity 



Hence the modulus is ^-w 2, » =#afo)X- ( - J n , the units a, fo being 



reintroduced to give the expression the proper dimensions. Now 

 </afo is a pressure, and therefore the length of the modulus is 



\W n ' ^"k us * ae sur f ace matter has a length modulus equal to a/w. 



Now let us find the ellipticity of the internal strata. 

 Substituting for w from (9) in (2), we have, 



a— + 2(3-n)— -2n-=0. 



da 2 da a 



If the solution be assumed of the form e=ca&, y3 must satisfy 

 /3(/3— l)+2(3— 2»=0, 

 whence /3 = — ( f - n) ± j {(f) 2 — n(3 —n) } . 



