1902-3.] 
Notes on Mr G. Romanes Paper. 
421 
those just indicated. He also found that when Laplace’s^ 
formula * for the variation of density with depth is used, the 
results differ but little from those obtained when the linear law 
of change adopted by Mr Romanes is made the basis of the- 
calculation. 
The results of the mathematical discussion are as follows^ 
With the assumption of the linear law of density, namely— 
Px = P + c(a-x); 
where p is the value at the surface, a the earth’s radius, x the- 
distance of the point considered from the centre, and c a con- 
stant, which can be determined in terms of the mean and surface- 
densities, the expression for the exhaustion of potential energy in 
building up a sphere of radius, a, by matter brought from infinity 
is — 
where h is the gravitation constant ; and the rate of change of this 
quantity per unit decrease of radius, the mass and p being 
constant, is — 
With a = 6*367 x 10 8 cms, p = 2'76, ac= 11*04, *=6*66 x ICR 8 in 
C.G.S. units, the values of E and - dE/da are — 
= 2*498 x 10 39 in ergs, - dE/da = 5*117 x 10 30 in ergs per cm. 
Now the earth loses about 70 calories per sq. cm. in one year, 
that is 3*57 x 10 20 from the whole surface, which is equivalent to- 
1*5 x 10 28 ergs. Thus a contraction of 1 centimetre would supply 
the heat radiated in 340 years ; or a contraction of 1 foot would 
supply the heat radiated in 10,000 years. 
Professor Gray concludes with these sentences : “ That a 
contraction of the order of magnitude of this small amount does 
take place is, in my opinion, not impossible. A great part of the 
material of the earth is no doubt in a plastic state, but a good deal 
* Not much importance can be given to Laplace’s law above any other. 
Wiechert supposes that the earth consists of an iron central portion of nearly 
uniform density and a radius of about four-fifths of the earth’s radius. See 
G. Darwin’s paper on the “Theory of the Figure of the Earth” ( Monthly 
Notices , R.A.S., vol. lx., 189S). 
dE _ 2t r 2 7r 
da 315 ' 
^(252oV + 130a 5 cp + 13 a 6 c 2 ). 
0 1 0 
