207, 208] Binary Stars 229 
value decreases. Roche*, treating equations (206*8) and (206*9) by a laborious 
method of numerical calculation, found that there is only one maximum on 
each series. On the series SBJ (p = qo ) the maximum occurs as we have seen 
at B ; the value of w 2 /2Tr'yp here is 018712. Roche calculated the maxima 
of &) 2 /27r7/> on other series. On the series p = 0, the series of configurations 
in which the primary is infinitesimal, he finds the maximum value of 
ft) 2 /27T7p to be 0*046, and the configuration at which this maximum occurs is 
that in which a = l*63r 0 , b = '81r 0 ; this is represented by the point R" in the 
diagram. When p = 1, the maximum value of co^^iryp is 0*072, and Roche 
finds that the value of &> 2 /27r7 p at the various maxima increases continuously 
from p — 0 to p — qo . 
On connecting the points B, R", T" by a continuous line, we obtain the 
points at which ® 2 is a maximum on the various linear series. 
Stability. 
208. In a physical problem in which afl'hryp increased continuously, it 
would follow, from the principles already discussed, that all configurations on 
the left of the line BR"T" would be stable, while all the configurations on 
the right would be unstable. The configurations on the left would of course 
only have been proved stable so long as the configurations were constrained 
to remain ellipsoidal, but it can be proved that this restriction makes no 
difference *f*. 
In the natural double-star problem, the change in physical conditions is 
not adequately represented by making &r/27T7p increase continuously. Even 
* l.c. p. 251. + Problems of Cosmogony, p. 85.