M. Poinsot on the Percussion of Bodies. 287 



;quations with a vie\ 

 ctions oi X, y, a and i 



Combining these two equations with a view of obtaining the 

 values of t and u as functions of x, y, a and b, we find 



t = i 



u^ax+^by-u^x^-^y 



a.' 



in the same manner, without calculation, and by simply inter- 

 changing X and y, a and b, a and /3, we have 

 _ ^ ^Hy-b)+x{ay-bx) 

 " ~ "^ u^ax + ^by - ct^x' -jB^y^ ' 

 Either of these two expressions, however, will here suffice ; for 

 since 



Q-P ^ 

 ^~ DO' 



and the ratio of the two lines C and D is the same as that 

 of their projections upon either of the two coordinate axes, we 

 have, by projecting upon the abscissa axis, the simple expression 



x — t 

 Substituting for t its value in x and y, as above given, we find, 

 after necessary reductions, the elegant formula 



This, then, is the expression for the force Q with which a body, 

 animated by an impulse P perpendicular to one of its principal 

 planes, is able to strike an obstacle presented at any point D in 

 the same plane. 



Corollary I. 

 19. In the first place, it will be seen that this percussion Q 

 is zero at all points for which 



that is to say, at all points of the spontaneous axis S corre- 

 sponding to the centre of impulsion C. 



In the next place, it is clear that this percussion Q will be 

 equal to the force P itself when 



« V + ^y'^ = c^'^ax + ^by ; 



whence it follows that the body strikes with the same force P, 

 not only at its centre of impulsion C, and at its centre of gravity 

 G, but also at all points of the circumference of the ellipse de- 

 scribed upon the line C G, with two axes parallel and propor- 

 tional to the axes a. and /3 of the central ellipse of the body. 



