158 



Mr. P. H. Ling on a Certain 



repeat Poincare's theorem, when the restriction as to motion 

 in one plane is removed. The result seems sufficiently un- 

 expected to be worth recording, for it appears to indicate 

 the possibility of a second integral. The present paper also 

 contains a number of inferences respecting this integral, 

 which, as need hardly be said, would be of some importance 

 in astronomy. 



1. The Equations of the System. 

 Let be the centre of gravity of S and J. Let Ox be an 

 axis coincident with S J, Oy a perpendicular one in the plane 

 of motion of S and J, and Oz one perpendicular to that 



plane. Let M, fi be the masses of S and J respectively, 

 n the angular velocity of the line SJ. Let OS = a, OJ=fr, 

 p 1= SP, p 2 =PJ. U the potential at P due to S and J. 

 Then 



Pi p2 



Also, if SJ is taken of unit length, 



n' 2 = M + yu, and Ma=fjLb. 

 Then the equations of motion are 



x — 2ny = n z x -f ^r— = -=r — 

 ox ox 



y + 2n f x = n r2 y + 





By 



3* 



an 



9y 



y, 



(i) 



where 



Pi P2 



