254 The Evolution of Binary and Multiple Stars [CH. xi 



the arrangement will be of the type represented in fig. 44. In such a con- 

 figuration, each mass will exert a couple on the other in such a direction as 

 to augment the motion of revolution already taking place. These couples 



Fig. 44. 



are the direct successors of the forces of restitution, mentioned in 264, 

 which tend to equalise the periods of rotation and of revolution. Let us 

 investigate the effect of the couples on the orbits of the masses. 



Suppose the original star of mass M + M ' to divide into two components 

 of masses M, M', each of which will describe an approximately elliptic orbit 

 about the centre of gravity of the two. Let e be the eccentricity and a the 

 semi-major-axis of the orbit of 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, 



-k, where A .-|^.(l-,) ......... (579), 



Energy = E, where E^-MM'fta .................. (580). 



Let the couples produced by tidal friction be supposed to act for a short 

 interval dt, each star exerting a couple G on the other in the direction of 

 rotation. 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 + ddt, in which e and a may be regarded as rates of 

 increase during the action of the couple G. 



These rates of change are readily found. From equation (580) 



1 2E 



8othat 



a~ MM" 

 1 da 2 dE 



Since G must be supposed to act in the direction of 6 increasing, GO will 

 be positive, so that da/dt is positive. Tidal friction increases a. 



By logarithmic differentiation of equation (579) 



1 d_ ox_2<^_lc7a 



l- e *dt ( ~hdt adi 



