22 Mr. J. C. Glashan on the Motion of a Particle 



the origin, and then varying only while the tensor is nil; in fact 

 here neither can vary with the other. Again, an infinitesimal 

 transverse "aphelion" motion will produce a relative-infinite 

 transverse " perihelion " motion ; but if the transverse " aphe- 

 lion " motion vanish absolutely, the " perihelion " motion does 

 so too ; but it would require to be absolutely infinite were rec- 

 tilinear motion the limiting case of motion in a fixed ellipse. 



The truth of the first statement in (2) appears from the above, 

 since x varies independently of the versor. The equation defines 

 the motion of a particle attracted while on the positive side, but 

 repelled on the negative side of the origin. In fact the force is 

 converted by each passage of the particle through the origin. 



The equation in (3) may be obtained thus : — Writing p (a 

 tensor) for the length- ratio of the radius vector, i a (primitive) 

 27r-root of unity for the versor, and n an integer, we have ge- 

 nerally 



(IP ~ ~f~ W 



If 6 be variable this will give n constant and 



_ I 



"~ 1 — ecos {6 —a) ' 



in which Z is a scalar. The path of the attracted pa/ticle will 

 be (pt mr+0_a ), a conic section. 



If 6 be constant, say =/3, (A) becomes ° 



\P). 



df ~ p 2 

 which reduces to 



d 2 x fju 



(C) 



dl* x( + \/x') 



Applying Professor Cayley's solution to (B),- we get 

 p = fl(l — cos</>), 



. t\Z/LL 



6—sm(p= — pi 



a* 



and the path of the particle to be pi nn +P. This path, interpreted 

 in the usual way for a radius vector passing through the origin, 

 gives for the particle an oscillatory motion through the centre 

 of attraction ; n will here be a discontinuous variable. 



I have used the above forms of (A) and (B) for the sake of 

 clearness in the dynamics; but analytically there is neither break 

 nor distinction. 



