from Rest towards an Attracting Centre. 23 



~a¥~~ f~* {U) 



.•. p{I—ecos (6—a)}=I, 



in which / may contain the versor i nK . The figure" is pc nir+& ~ a 9 

 a conic section or straight line according as I is actual or nil. 

 Taking the case of the ellipse and tracing its change of form as 

 / varies from positive through zero to negative, we have, fig. 1, 

 / finite and positive ; fig. 2, limit to which fig. 1 tends as I ap- 

 proaches zero ; fig. 3, limit from which fig. 4 tends as I draws 

 away from zero negatively - } fig. 4, I finite and negative. The 



Fig. 1. 



Fiar. 2 a. 



A 



a* it 



Fi 2 . 3. 



Fig. 4. 



.\' 



•^ 



passage from fig. 2 to fig. 3 occurs discontinuously at p = 0. 

 (In the plane sections of the cone, figs. 2 and 3 occur simulta- 

 neously, so that a line from A to A! appears in the primary 

 plane, and one from a to a! in the secondary. There is no dis- 

 continuity in the motion of the section-plane, the analytical 

 discontinuity merely indicating that the passage from the lower 

 to the upper semi-cone is instantaneous, but through a position 

 in which both the semi-cones are cut.) 



For the particle to change from an ?i-even to an n-odd line 

 would require discontinuity of motion ; but in the problem there 

 is no force to cause this discontinuity ; for it would needs occur 

 at the centre, and there the attraction is nil. The motion will 



