49-52] The Double-Star Problem 47 



ROCHE'S PROBLEM 



51. The simplest problem occurs when the secondary may be treated as 

 a rigid sphere ; this is the special problem dealt with by Roche. As in 47 

 the tide-generating potential acting on the primary may be supposed to be 



We shall for the present be content to omit all terms beyond those written 

 down. The correction required by the neglect of these terms will be discussed 

 later, and will be found to.be so small that the results now to be obtained are 

 hardly affected. 



On omitting these terms, and combining the two potentials (87) and (88), 

 it appears that the primary may be supposed influenced by a statical field of 

 potential 



The terms in x may immediately be removed by supposing o> to have the 

 appropriate value given by 



M+M' 



and the condition for equilibrium is now seen to be that we shall have, at every 

 point of the surface, 



F 6 +^(^-J2/ 2 -J^)-i-|a> 2 (^ + 3/ 2 ) = cons ............. (90), 



where ft again stands for M'/R S . 



52. Equating the left-hand of equation (90), as before, to 



we find, as the conditions of equilibrium, 





. _ 



A Trpabc ^Trpabc a? 



T > 



> 



(91) 



It will be seen that these equations are more general than either of the 

 two sets we have considered before, each of which are indeed included as 

 special cases in the present set. Putting fju = we obtain the equations of the 

 rotational problem, while on putting w = we obtain the equations of the 

 tidal problem. 



