276 M. Poinsot on the Percussion of Bodies. 



as well as their possible use in mechanics justify, we think, all 

 the details and all the developments we have given in the fore- 

 going memoir. 



But these elegant propositions are themselves merely corol- 

 laries of other more general ones, as we shall see in the follow- 

 ing chapter. 



Chapter II. 



§1- 



1. Hitherto we have supposed the direction of the actual im- 

 pulse animating the body to be in the plane of its two principal 

 axes GX and GY, so that spontaneous rotation took place 

 around an axis parallel to the third principal axis G Z. 



We will now consider the case where the impvdse P is given 

 in a direction perpendicular to the plane of the two axes G X 

 and G Y, and at any point C t 



in this plane, which point we 

 shall call the centre of impul- 

 sion. Such an impulse gives 

 rise, as we shall show, to a _ 

 spontaneous axis in the plane 

 containing these two principal 

 axes ; the point 0, where the 

 line C G produced meets this 

 spontaneous axis S, will be referred to as the spontaneous 

 centre corresponding to the centre C. As will be seen, these 

 two centres are always reciprocal; that is to say, if the im- 

 pulse were given at 0, a spontaneous centre would be formed 

 at C, and the spontaneous axis C T would be parallel to the 

 first O S. 



The question to be first solved therefore is the following : — 



Problem I. 



Given, in the plane of two principal axes of a body, the centre 

 C of an impulse P perpendicular to this plane, to determine the 

 corresponding spontaneous axis O S. 



Solution. — Let x and y be the coordinates of the point C, where 

 the impulse is applied, with respect to the two principal axes G X 

 and GY under consideration; let m be the mass of the body, 

 and ma.^, mjS^ its moments of inertia around the axes G X and 

 G Y respectively. 



If P be transported parallel to itself from C to C on the axis 

 G X, C being the foot of the ordinate y of the point C (see fore- 

 going figure), we shall have in the first place a couple, with the 

 moment Py, tending to produce rotation around the axis G X 



