52, 53] The Double-Star Problem 49 



locus p = oo (Maclaurin spheroids), so that it is impossible for any of the loci 

 ever to cross this line ; they all lie completely on one side or the other of it. 

 Moreover the values of p and //, must obviously vary continuously as we move 

 continuously in the a, b plane. 



In fig. 7 such a diagram is represented. The point S(a=b = r ) represents 

 the spherical configuration ; the line OSM (a = b) is the series of Maclaurin 

 spheroids, and the line TST' (ab 2 = 1) is the series of tidal spheroids. B is the 

 point of bifurcation on the series of Maclaurin spheroids and JBJ' is the series 

 of Jacobian ellipsoids. 



All points which are on the side T'J' of the median line OSM represent 

 configurations for which b > a, and therefore configurations in which the pri- 

 mary is broadside on to the secondary. It is obvious that all these configurations 

 are unstable, for they would be unstable even if the primary were constrained 

 to remain rigid. These configurations need not trouble us further and we may 

 confine our attention to the right-hand half of the diagram. 



Linear series for all values of p pass through 8. The series for p = + oo is 

 the broken line SBJ, that for p = 1 is the line ST, while that for p = oo is 

 the line SO. Remembering that two linear series cannot cross, it is clear that 

 the series for a very large positive value of p must be asymptotic to the line 

 SBJ. All the series from p = + cc to p = 1 accordingly "lie within the small 

 area bounded by the lines JB, BS, ST. The series in the area OS T are of 

 course series for which p is negative and numerically greater than 1, while 

 those in the area MBJ are again series for which p is negative, a second 

 series for p = oo coinciding with the line MBJ. 



Let us now confine our attention to the series which lie inside th,e area 

 JBST, these being as we have seen the only ones of physical interest. Each 

 series starts at S and ends at the point in which the lines BJand ST ultimately 

 meet at infinity. Thus each series begins with a sphere and ends with an in- 

 finitely long prolate spheroid. As we pass along any one of these series /* 

 changes while p remains constant. The value of o> 2 , which is given by equation 

 (94) accordingly changes also, this giving the value of a real angular velocity 

 when p is positive, and being regarded simply as an algebraic quantity when 

 p is negative. The value of &> 2 vanishes only when p = 1 or when //. = ; 

 consequently it vanishes at S, at (JT)^ and along the line ST. It follows that 

 ft> 2 is of the same sign everywhere inside the area SBJT, and this sign is readily 

 seen to be positive. 



Since a> 2 vanishes at both the ends $ and (JT)^ of every series, it follows 

 that on passing along each series o> 2 at first increases, and then after passing 

 a maximum value decreases. Roche*, treating equations (98) and (99) by a 



* I.e. p. 251. 

 j. c. 4 



