53-55] The Double- Star Problem 51 



laborious method of numerical calculation, found that there is only one maxi- 

 mum on each series. The maximum on the series SBJ (p oo ) occurs as we 

 have seen at B; the maximum value of G) 2 /27rp is 0*18712. Similarly the 

 maximum on the series ST(p = 1) occurs when /-t is a maximum^nd so at 

 the configuration of eccentricity '8826. This is represented by the point T" 

 in the diagram, the value of a) 2 /2?r / o at this point is 0. Roche has calculated 

 the maxima of a) 2 /'27rp on other series. On the series p = 0, the series of con- 

 figurations in which the primary is infinitesimal, he finds the maximum value 

 of co' 2 /27rp to be O046, and the configuration at which this maximum occurs is 

 that in which a = I'63r , b = -81r ; this is represented by the point R" in the 

 diagram. When p = l, the maximum value of &> 2 /27rp is O072, and Roche 

 finds that the value of this maximum increases continuously from p = to 

 p = oo . 



On connecting the points B, R", T" by a continuous line, we get the loci 

 of points at which <w 2 is a maximum on the various linear series. 



Stability 



54. In a physical problem in which &> 2 increased continuously, it would 

 follow, from the principles already discussed, that all configurations on the left 

 of this line would be stable, and all the configurations on the right would be 

 unstable. The stability of the configurations on the left would of course only 

 be stability so long as the configurations were constrained to remain ellipsoidal, 

 although we shall see later that this restriction makes no difference. 



In the natural double-star problem, the change in physical conditions is 

 not represented by an increase in <u 2 . Both masses lose energy by radiation, 

 and shrink accordingly. The rates of shrinkage, and consequent rates of in- 

 crease in density will in all probability be quite different for the two masses. 

 We can, however, construct an artificial problem in which the density, if sup- 

 posed uniform to begin with, remains, uniform throughout the shrinkage, or 

 in which the two densities, if not supposed equal to begin with, change so as 

 always to retain the same ratio. The physical conditions are now represented 

 by an increase in the absolute densities, while the moment of momentum re- 

 mains constant and, exactly as in 44, these conditions may equally be 

 represented by supposing both densities to remain constant while the moment 

 of momentum increases. 



55. Let us first consider the general problem in which the secondary is 

 not regarded merely as a point. 



The moment of momentum of the primary about the centre of gravity of 

 the system is 



42 



