160-162] Roche's Model 161 



where V M is the potential of the spheroidal primary, and p stands as usual for 

 M'/R 3 . The point of equilibrium on the #-axis is determined by 9n/3# = 

 or (cf. 154) 



The integral can be evaluated in finite terms, so that the equation becomes 



where a 2 = a 2 - c 2 . 



In the special case in which the nucleus of the primary is on the verge 

 of instability, the value of /j, is 0'1255047r/) , while 



a == (a 2 - c 2 )^ = 1-45970 (a&c)*. 

 Thus equation (421) reduces to 



log 5-^-? = -390343, 



x a. x 



of which a root is readily found to be 



x= 1-37578 a = 2*00822 r . 



Fig. 32. 



The corresponding figure of equilibrium is shewn in fig. 32. The thick 

 curve is the boundary of the nucleus, and the thin outer curve that of the 

 greatest atmosphere which can be retained by this nucleus. The volume V A 

 of this atmosphere is only about a tenth of 0y the volume of the nucleus. Thus 



- ^ and B- 1-70 r.. 



Hence we arrive at the following conception of the series of configurations 

 of this model. At first, when the tidal forces are inappreciable, the figure of 

 equilibrium is spherical, this giving place to a spheroidal figure where the 

 tidal forces become appreciable but small. As the tidal forces increase, the 

 boundary of the nucleus remains spheroidal, but that of the atmosphere is a 

 pseudo-spheroid. If the volume of the atmosphere is greater than about a 

 tenth of that of the nucleus, this pseudo-spheroid develops two conical pointed 

 ends at the extremities of its major-axis, and a further increase in the tidal 

 forces results in matter streaming out from these two ends. (This is in the 

 j. c. 11 



