EVOLUTION OF THE SOLAR SYSTEM. 
515 
We may here remark that the secular effects of tidal friction in the case of a rigid 
sun attended by tidally-disturbed planets, with no satellites, may easily be determined. 
For if we put c i =x, and note that SI varies as x~'\ and that n has the form (h — Jcx) / C', 
we see that it would, only be necessary to evaluate a series of integrals of the form 
-"t " t . This integral is in fact merely the time which elapses whilst x changes 
J x a °- —px V 'yx 
from x 0 to x, and the time scale is the same for all the planets. It is not at present 
worth while to pursue this hypothetical case further. 
Now if we suppose the planet to raise frictional tides in the sun, as well as the sun 
to raise tides in the planets, we easily see by a double application of (2) that 
, Mm dc} _. 2 1 
H ‘\m+ m)* dt = ^ ? 
(28) 
The tides raised in the planet by its satellites do not occur explicitly in this equation, 
but they do occur implicitly, because n, the planet’s rotation, is affected by these tides. 
The question which we now have to ask is whether in the equation (28) the solar 
term (without accents) or the planetary term (with accents) is the more important. 
In the solar system the rotations of the sun and planets are rapid compared with the 
orbital motions, so that S2 may be neglected compared with both n and ri. 
Hence the planetary term bears to the solar term approximately the ratio 
JFCV qp 
m 2 6 r wg / p' 
Now 
(*)'?*. AIso wqv. 
\mj (J g m\aj g a \g j a p \g j v 
Therefore the ratio is 
Now solar gravity is about 26‘4 times that of the earth and about 10‘4 times that of 
Jupiter. The solar radius is about 109 times that of the earth and about 10 times 
that of Jupiter. The earth’s rotation is about 25 '4 times that of the sun, and Jupiter’s 
rotation is about 61 times that of the sun. Combining these data I find that the effect 
of solar tides in the earth is about 113,000 v/v times as great as the effect of terrestrial 
tides in the sun, and the effect of solar tides in Jupiter is about 70,000 v/v times as 
great as the effect of Jovian tides in the sun. It is not worth while to make a similar 
comparison for any of the other planets. 
Now it seems reasonable to suppose that the coefficient of tidal friction in the planets 
is of the same order of magnitude as in the sun, so that it is improbable that v/v 
should be either a large number or a small fraction. 
We may conclude then from this comparison that the effects of tides raised in the 
sun by the planets are quite insignificant in comparison with those of tides raised in 
the planets by the sun. 
It appears therefore that we may fairly leave out of account the tides raised in the 
3 x 
ig \ 4 a' n' v' 
Vj') a n v 
MDCCCLXXXI. 
