EVOLUTION OF THE SOLAR SYSTEM. 
513 
in the sun by the planet both go towards expanding the orbit of the planet. It is this 
latter effect with which we are at present concerned. 
I propose then to consider the probable relative importance of these two causes of 
change in the planetary orbits. 
But before doing so it will be well as a preliminary to consider another point. 
In considering the effects of tidal friction the theory has been throughout adopted 
that the tidally-disturbed body is homogeneous and viscous. Now we know that the 
planets are not homogeneous, and it seems not improbable that the tidally-disturbed 
parts will be principally more or less superficial—as indeed we know that they are in 
the case of terrestrial oceans. The question then arises as to the extent of error 
introduced by the hypothesis of homogeneity. 
For a homogeneous viscous planet we have shown that the tidal frictional couple is 
approximately equal " to 
n—fl 
. aavj 
where |)=bu — 
19u 
Now how will this expression be modified, if the tidally-disturbed parts are more or 
less superficial, and of less than the mean density of the planet ? 
To answer this query we must refer back to the manner in which the expression was 
built up. 
By reference to my paper “On the Tides of a Viscous Spheroid” (Phil. Trans., 
Part I., 1879, pp. 8-10, especially the middle of p. 8), it will be seen that jj is really 
(%gciw—%gaiv)/l9v, and that in both of these terms iv represents the density of the 
tidally-disturbed matter, but that in the former g represents the gravitation of the 
planet and in the second it is equal to ^irgaiv, where w is the density of the tidally- 
disturbed matter. Now let f be the ratio of the mean density of the spheroid to the 
density of the tidally-disturbed matter. 
Then in the former term 
And in the latter 
gaiv— 
kirix, 
g . %Triiaw=—g 2 X 
1 
/ 
gaw— 
1^ 
P 
Hence if the planet be heterogeneous and the tidally-disturbed matter superficial, p 
must be a coefficient of the form 
_3_ f /5_3\ 
4z7Tfj, x 19 vf \2 2 f) 
If f be unity this reduces to the form gawj\9v, as it ought; but if the tidally- 
* I leave out of account the case of “ large ” viscosity, because as shown in a previous paper that could 
only be true of a planet which in ordinary parlance would be called a solid of great rigidity.—See 
“Precession,” Phil. Trans., 1879, Part II., p. 531. 
