280 M. Poinsot on the Percussion of Bodies. 



diameter 8' which is parallel to the spontaneous axis OS, we 

 shall have 



8' ~ 8' 

 but in the ellipse 



«y8 = SS' sin (f>, 



where (j) is the angle between the conjugate diameters 8 and 8' ; 

 hence 



K = S sin <^ 



is the value of the arm of inertia around the diameter conjugate 

 to 8, and it is also the distance of the extremity of the latter dia- 

 meter from its conjugate 8'. 

 The above equation, 



therefore, becomes 



A sin <f). H sin (^ = K^, 



and shows that the distances A sin ^ and H sin cf) of the reci- 

 procal centres C and from the diameter GD', parallel to OS, 

 are such that their product is equal to the square of the arm of 

 inertia of the body around this diameter GD', This proposition 

 is exactly similar to the one demonstrated in the first chapter 

 (art. 5) ; it is, however, more general than the latter, and con- 

 tains it as a particular case. 



Corollary I. 



6. We have just seen that if a; and j/ are the coordinates of 

 any point C, considered as a centre of impulsion, the equation of 

 the spontaneous axis corresponding to this centre C will be 



cc^xt + /3hju + u^f3^ = 0, 



where t and u are the current coordinates. It is manifest that if 

 a; and y vary, in other words, if we suppose C to change its place, 

 the line OS will change its position also. It might be asked, 

 therefore, what relation ought to be established between x and 

 y in order that the spontaneous axis OS may always pass through 

 one and the same point 0, having the coordinates t' and u'. To 

 find this relation it is evidently sufficient to regard the preceding 

 equation as being always satisfied for the coordinates / = /',« = «' 

 of the given point ; the required relation between x and y, 

 therefore, is 



which we at once recognize to be the equation of a spontaneous 

 axis CT corresponding to the point considered as a ceuti'e of 

 impulsion. 



Thus, in order that the several centres of inq)ulsion C, C, C", 



