284 



M. Poinsot on the Percussion of Bodies. 



and y being, as usual, the coordinates of the point C, and / and 

 u those of the reciprocal centre 0, we have, between the coordi- 

 nates of these two points, the equations 



y=- 





If, therefore, the centres C being taken on any curve whatever 

 given by the equation y—f(^\ 



the locus of the corresponding centres O be required, it is only 

 necessary to substitute, for x and y, the preceding values in order 

 to obtain immediately, between t and u, the equation of the 

 required locus of the centres 0. 



And conversely, given the locus of the spontaneous centres, 

 that of the centres C would be found by means of the expres- 

 sions reciprocal to the foregoing. 



15. Let the centres C be situated in any given right line; the 

 latter may always be represented by an equation of the form 

 o?ax + ^by-\-u^^=0, 



provided we give suitable values to the two arbitrary coefficients 

 a and b, and if in this equation we express x and y in terms of 

 t and u, we shall have, for the locus of the corresponding spon- 

 taneous centres 0, the equation 



uH'^ + /8V _ a?at-^%u = 0. 

 This is an ellipse similar and similarly placed to the central 

 ellipse; its centre, however, is at the point I, whose coordinates 



are ^ ^^d s? and its circumference passes through the origin G. 



