227 
гое, 207] Binary Stars 
and, on equating coefficients of x 2 , y 3 and z 1 , we obtain as the conditions of 
equilibrium, 
j P - (206-61), 
A тгураЬс 2irypabc a? 
jt j +_£ ^£ (206-62), 
^ в 2 г тгураЪс 2тгураЬс b 2 
J c + —l Ц-=Л (206-63), 
2тгураЬс с 2 
where p is written for yM'/R 3 . Putting p = 0 causes the secondary to go out 
of existence. The problem then reduces to that of a single mass rotating 
freely in space, the equations becoming identical with those already discussed. 
It is convenient to put M/M' =p, so that 
a>* = (l+p)p (206-64). 
The equations then reduce to 
(3±pb 0 (206-71), 
2’rrypabc a 
л-к£в-ё (206 ' 72) ’ 
'• + h£k-? ; (20673) - 
The elimination of в from the first two equations gives 
( b2 “ io (b* + \) (c 2 + X) A = ( b P + c ) 2tt ypabc 
(206-8), 
while similarly the elimination of в from the first and third yields 
(ft + X) ? + X) д = ^ + c ' + ba?) 
(206-9). 
Except for differences of notation, these two equations are identical with 
the two which Roche takes as the basis of his discussion*. 
207. It will be most convenient to examine the solution of these equations 
by a graphic method. We may suppose a, b, c to be connected by the usual 
relation abc = r 0 3 , where r 0 remains constant throughout the changes of shape 
of a given mass. Thus the two quantities a, b specify the shape of the mass. 
Let us take a and b as abscissa and ordinate, as in fig. 33. At each point 
in this diagram the values of p and p may be uniquely determined from 
equations (206*8) and (206*9). If we map out a curve along which p or M/1W 
is constant, this curve will represent a linear series of configurations correspond 
ing to different values of p or yM /R 3 , and so corresponding to different 
* Acad, de Montpellier (Sciences), i. (1850), p. 243. Our two equations (206 - 8) and (206 - 9) are 
identical with equations (4) and (5) (p. 247) of Roche’s memoir.