M. Poinsot on the Percussion of Bodies. 283 



These two values being substituted in the preceding equations 

 (1) and (2) and x subsequently eliminated, there will result, be- 

 tween t and u, an equation 



which will give the required curve formed by the successive in- 

 tersections of the spontaneous axes, and to which these axes are 

 tangents. 



11. For example, if the given locus of the centres be a para- 

 bola 



the required curve will be found to be 



which is also a parabola whose parameter is inversely propor- 

 tional to the parameter A of the first ; so that if A were given 



equal to 2 — , the two parabolse would form but one, — a result 

 ot, 



in pei-fect accordance with what we have seen in the preceding 



corollary. 



12. If the centres were taken along the contour of the cen- 

 tral ellipse whose equation is 



the curve enveloped by the corresponding spontaneous axes 

 would be found to be 



or the same ellipse ; in fact since, by hypothesis, each centre is 

 at the extremity of a diameter, the corresponding spontaneous 

 axis is nothing more than the tangent at the other extremity. 



13. If the locus of the centres were the circumference of a 

 circle given by the equation 



x'' + y'' = ^\ 



the equation of the required curve would be found to be 



which is that of an ellipse having the semi-axes -5-, -^ directed, 



respectively, along the axes /3, a of the central ellipse, but pro- 

 portional in length to the squares of these axes. 



Corollary V. 



14. Questions relating to the mutual positions of the cen- 

 tres C and O of percussion and spontaneous rotation arc still 

 simpler, and more easily solved than the preceding ones ; for x 



