12 Proceedings of the Royal Irish Academy. 



secures that a stable circular orbit shall be converted into a stable orbit, 

 and an unstable into an unstable. Among the problems soluble in terms 

 of circular functions, we have p = - 2, k = 2, q = 1, which is the case 

 already examined in detail. The remaining case in this class is p = - 3, 

 corresponding to which q becomes infinite. Here it is readily seen that 

 the result of making JV constant is to convert the problem into that of 

 a parallel field of force in which the potential is of the form /ne c K 



7. In the problem of two fixed centres of gravitation 



V = Ml {{x - cf + f}-^ + ^ {(x + cf + y*)' 1 



Hence 



v , = /,' . /,' (m IfiA +/ic +/.c * c^ | 



~ (fi'-tf (A 2 -rf (+ m(Jif,-fx*-M + <*)*-A(/, , -<0*C/i , -«0*J' 



This naturally suggests writing 



f = c cos $, 

 whereby V becomes 



V = (p'j <j>\ [2fi,c cos l^ cos |02 + 2ju 2 c sin |^, sin |^ 2 - c 2 A sin 0i sin $ 2 | . 

 Taking <p to be the simple linear function £ + itj, we then get 



F' = fiiC (cosh »j + cos £) + ;u 2 c (cosh »; - cos £) - \drh (cosh 2»j - cos 2?), 

 which leads to the equations 



-j-j •= (ju» - jui) c sin £ - c'A, sin 2?, 

 -3-; = (^2 + Hi) c sinh »/ - c a A sinh 2^, 



and t = \ Jdr = |c 2 J (cosh 2r\ - cos 2§) At. 



This is the ordinary solution, £ and ») being expressible as elliptic functions 



of r. (Cf. Whittaker, Analytical Dynamics, p. 95.) 



8. In the simplest case of the problem of three bodies, the restricted 

 problem with equal finite masses, we still have with this transformation 



J = 3c 2 (cosh 2>> - cos 2£) ; 

 and it is necessary to add to V the term 



|nV(a? + y»), WA'A'fif* 



= £«V sin 20, sin 20 2 



= -jVj-V (cosh 4rj - cos 4?) . 



With unequal masses it would be necessary to remember that the origin of 



