EVOLUTION OF THE SOLAR SYSTEM. 
527 
In accordance with the above supposition h is a quantity which diminishes slowly in 
consequence of tidal friction, and a diminishes in consequence of contraction, at such 
a rate that dnjdt is positive. 
We also have 
m= 
2x3 
sr 
19 X 5 X 47r f.LVCJb 
The rate of change in the dimensions of the planet’s orbit about the sun remains 
insensible, so that r and fL may be treated as constant. 
t 2 / n 
Then the rate of loss of rotational momentum of the planetary mass is Cn—[ 1— 
!JP\ n 
, D,ma?\ 
*~7 
/ ^ 077 \ ^ 
w~K Also g=jxm/a 2 . 
By the above transformations we see that this expression varies as ^ 
/ Q'TY) \ ^ 
But m=^7rwa s , and therefore a 
1-f 
\47T 
On substituting this expression for g, and then replacing a throughout by its expres¬ 
sion in terms of iv, we see that, on omitting constant factors, the rate of loss of 
• 1 • Jlv JbV , 7 9 / 3 , 
rotational momentum varies as —i-- where A'= 4 - S2m\ a constant. 
w 3 vr 0 \47r/ 
From (29) we see that if h varies as a 3 , or as iv~*, n the angular velocity of rotation 
remains constant. 
If therefore we suppose h to vary as some power of a less than 2 (which power may 
vary from time to time) we represent the hypothesis that the contraction causes more 
acceleration of rotation than tidal friction causes retardation. Let us suppose then 
that li =ELc - ^' 3 where /3 is less than ■§-, and varies from time to time. 
Then the rate of loss of momentum varies as 
^- 3 (H v?-k). 
In order to determine the rate of loss of rotation we must divide this expression 
by C, which varies as iv~K 
Therefore the rate of loss of angular velocity of rotation varies as 
In order to determine how tidal friction and contraction co-operate it is necessary to 
adopt some hypothesis concerning the coefficient of friction v. 
So long as the tides consist of a bodily distortion, the coefficient of friction must be 
some function of the density, and will certainly increase as the density increases. 
Now if, as regards the loss of momentum, v varies as a power less than the cube of 
the density, the first factor vio~ 3 diminishes as the density increases ; and if, as regards 
the loss of rotation, v varies as a power less than % of the density, the first factor vw~* 
diminishes as the density increases. 
And as the cube and | powers both represent very rapid increments of the coefficient 
of friction with increase of density, it is probable that the first factor in both expressions 
diminishes as the contraction of the planetary mass proceeds. 
Now consider the second factor H :uf — k, which corresponds to the factor n — fl in the 
