514 
ME. G. H. DARWIN ON THE 
disturbed matter be superficial and of less than the mean density, then jp must be 
«2 / g\ 
a coefficient wbicb varies as —4 1—— . The exact form of the coefficient will of 
H’Hf\ 5 fJ 
course depend upon the exact nature of the tides. If f be large the term 3/5 f will be 
negligeable compared with unity. Again, if we refer to the following paper 
(“Precession, &c.,” Phil. Trans., Part II., 1 879, p. 456), it will be seen that the C in 
the expression for the tidal frictional couple represents ■§-(■§7 ra?w)a z , where w is the 
density of the tidally-disturbed matter; hence C should be replaced by C/f. 
Then if we reconstruct the expression for the tidal frictional couple, we see that it is 
to be divided by f because of the true meaning to be assigned to C, but is to be 
multiplied by f on account of the true meaning to be assigned to p. 
From this it follows that for a given viscosity it is, roughly speaking, probable that 
the tidal frictional couple will be nearly the same as though the planet were homo¬ 
geneous. The above has been stated in an analytical form, but in physical language 
the reason is because the lagging of the tide will be augmented by the deficiency of 
density of the tidally-disturbed matter in about the same proportion as the frictional 
couple is diminished by the deficiency of density of the tide-wave upon which the 
disturbing satellite has to act. 
This discussion appeared necessary in order to show that the tidal frictional couple 
is of the same order of magnitude whether the planet be homogeneous or heterogeneous, 
and that we shall not be led into grave errors by discussing the theory of tidal friction 
on the hypothesis of the homogeneity of the tidally-disturbed bodies. 
We may now proceed to consider the double tidal action of a planet and the sun. 
Let us consider the particular homogeneous planet whose mass, distance from sun, 
and orbital angular velocity are to, c, fL. For this planet, let C — moment of inertia; 
a!= mean radius; w'= density; g'= gravity; g' = f g'ja ] v'= the viscosity; 
p'= g'a!w' /19t/ = 4 VT 9 </V 77 l JLV ' > ari( J- n '— angular velocity of diurnal rotation. 
The same symbols when unaccented are to represent the parallel quantities for the 
sun. 
Now suppose the sun to be either perfectly rigid, perfectly elastic, or perfectly fluid. 
Then mutatis mutandis, equation (2) gives the rate of increase of the planet’s distance 
from the sun under the influence of the tidal friction in the planet. It becomes 
i Mm dc h _, 3 .2 C fiuJf ) 2 n' — fl. 
^(M + my Q 7 ~~i* 
If the planet have no satellite the right-hand side is equal to —C'dn'/dt, because 
the equation was formed from the expression for the tidal frictional couple. 
Hence, if none of the planets had satellites we should have a series of equations of 
the form 
CV+/T 
Mm 
(M+m) h 
with different h ’s corresponding to each planet. 
r=h 
