294 
[ch. xi 
The Evolution of Binary Systems 
separate masses will gain on their revolution about one another, with the 
nature shewn in fig. 50. In a configuration such as this, each mass exerts a 
revolution already taking place. These couples are the direct successors of the 
forces of restitution, mentioned in § 255, which tend to equalise the periods of 
rotation and of revolution. The effect of these couples on the orbits of the 
261. Suppose that a star originally of mass M + M' has divided into two 
components of masses M y M', each of which describes an approximately 
elliptic orbit about the centre of gravity of the two. Let e be the eccen 
tricity and a the semi-major-axis of the orbit described by either mass 
relative to the other. 
If the tidal friction couples were non-existent, there would be the usual 
two first integrals of the motion, 
interval dt, each component of the system exerting a couple 0 on the other 
in the direction of the orbital motion. As a consequence, the orbit will be 
disturbed and at the end of the interval dt a new orbit will be described. 
The eccentricity and semi-major-axis of this may be denoted by e+edt, 
a + adt, in which e and a are regarded as rates of increase during the action 
of the couple G. These rates of change are readily found. Differentiating 
equation (261*3), we find 
Since G acts in the direction of 6 increasing, G6 must be positive. Thus 
da/dt is positive, so that tidal friction increases a. 
By logarithmic differentiation of equation (261*2), 
result that after a time the arrangement of the masses will be of the general 
couple on the other in such a direction as to augment the orbital motion of 
masses has been very fully investigated by Darwin under the designation of 
“ Tidal Friction.” 
Fig. 50. 
(261-1), 
where 
M^M"* 
h * = ^WTW' a (l“ e2) (261 ' 2) ’ 
Energy = E, where E = — yMM'/2a (261"3). 
Let the couples produced by tidal friction be supposed to act for a short 
.(261-2), 
1 d 
2 dh 1 da