M. Poinsot on the Percussion of Bodies. 285 



The right hue C T, having been represented by the equation 



may, manifestly, be regarded, as the spontaneous axis correspond- 

 ing to a centre whose coordinates are a and b. The centre I 

 of our eUipse, therefore, is the middle point of the line G O ; 

 and the ellipse itself is constructed upon G as diameter homo- 

 logous to the like-directed diameter in the central ellipse. 



16. Hence, regarding the given line CY as a spontaneous axis 

 corresponding to a centre 0, we may say that if the several points 

 of this line be taken as centres of percussion, the corresponding 

 spontaneous centres will be all situated on an ellipse similar to the 

 central ellipse, and described upon the line G as diameter homo- 

 logous to that which the direction of GO determines in the central 

 ellipse. And conversely, if the points of this ellipse were taken 

 as centres of percussion, the spontaneous centres would be all 

 situated along the line C T. 



On account of the reciprocity between the centres C and O, 

 it is evident that if the centres of percussion were the points of 

 the line S, parallel to C T, the corresponding spontaneous 

 centres would be on an ellipse similar to the first, but described 

 on C G as diameter homologous to G. 



17. As a second example, let the centres C be taken on the 

 parabola represented by the equation 



y''=Kx; 



putting in place of x and y their values in t and u, we find the 

 equation 



KuH^ + k^%H + «2y8 V = 0, 



which gives the required locus of the spontaneous centres cor- 

 responding to the centres C. 



Solved with respect to the ordinate u, this equation gives 



''=^W-a 



-A/3 



'■-\-Kt' 



whence, since a, /3, and A are positive, we see that this curve of 

 the third order has only real ordinates u on the side of the nega- 

 tive abscissjE t ; that, like the parabola {y^ = Aa;), the curve has 

 two equal branches which stretch to infinity above and below the 

 abscissa axis ; but that these two branches cannot recede from 

 the ordinate axis to a distance greater than 



^ A' 



so that a line drawn parallel to the ordinate axis, and at a di- 

 stance from the same equal to /', is an asymptote to both branches 

 of the curve in question. 



