160 



Prof. G. H. Darwin. 



[Dec. 13, 



Now if be a function such that wdp = dxs, the differentiation of 

 (6) leads to — 



d * + H a waUa=0 (7). 



da a^ J o 



and a second differentiation to — 



d? 



— -{ma)+4i7rwa=0 (8). 



da 2 



It is well known that Laplace assumed that the modulus of com- 

 pressibility of rock varies as the square of the density. Since this 

 modulus is ivdjpjdw, Laplace's hypothesis makes m proportional to w, 

 and the equation (8) is at once soluble. 



After the determination of w as a function of a, the solution of all 

 the other equations follows. 



In this paper I propose to find a new solution, and to compare the 

 results with those of Laplace. 



In order to simplify the analysis let the unit of length be equal 

 to the mean radius a of the planet, and the unit of time be such that 

 the surface density u) of the plant is also unity. 



Now let us assume that the law of internal density is — 



w—a~ n (9). 



Then the mean density of all the matter lying inside of the stratum 

 a is a~ n /(l—^n). Hence, by definition we have — 



k=l-in (10). 



Thus we see that Jc is a constant for all strata, and therefore also 

 for the surface. In Laplace's theory h is variable. With our assumed 

 law of density and the special units, p the mean density is equal to 

 the reciprocal of h. 



It is clear that n must be positive, otherwise heavier strata lie 

 above lighter, and it must be less than 3 in order to avoid infinite 

 mass at the centre of the planet. 



Now let us find the law connecting pressure and density, and the 

 modulus of compressibility. 



Equation (7) becomes 



da 3 



and by definition of w and the assumption (9), 



