282 M. Poinsot on the Percussion of Bodies, 



be a parabola having its vertex at G, GY as axis, and a parameter 

 equal to A. Our three equations will then be 



A»^x + /3^y.2t = 0; 



whence, eliminating i and u, we find that the centres C must be 

 situated on the parabola 



having the same axis and vertex as the given parabola, but a 



parameter A' = -t— g. 



Hence, when the centres of percussion are situated on this 

 parabola, all the corresponding spontaneous axes will be tangents 

 to the other given parabola. 



Provided A were equal to — — , then A' and A would be equal 



to one another, and the two parabolse would coincide ; so that the 

 centres being taken upon one branch, all the spontaneous axes 

 would be tangents to the other branch of the same parabola. 



Corollary IV. 

 10. The same analysis would lead us to the solution of this 

 inverse problem : given the curve which forms the locus of the 

 centres of percussion, to find the curve formed by the successive 

 intersections of the spontaneous axes, in other words, the curve 

 to which all such axes are tangents. 



For, X and y being still the coordinates of any one of the cen- 

 tres C, the corresponding spontaneous axis has the equation 



u^xt + /3^yu + u^^^ = (1) 



Now if an infinitesimal variation be given to x and y, t and u 

 being regarded as constant, we shall have 



u^t + ^u/£=0, (2) 



and these two equations between u and t are satisfied only at the 

 point of intersection of the two spontaneous axes corresponding 

 to the two centres infinitely close to each other. But, by hypo- 

 thesis, the locus of these centres is given, so that between x 

 and y we shall have a given equation 



whence we may deduce 



l=/'w- 



