OP THE MOTIONS OF A SOLID BODY. £6'9 



357. Theorem. In a homogeneous 

 sphere, the distance/, at which an impulse 

 must have been given, in order to cause a re- 

 volution and a rotation at once, must be I . ~* 



' r U 



R being the radius of the sphere, r its dis- 

 I tance from the centre of revolution, p the 

 angular velocity of rotation, and U that of 

 revolution. 



An impulse acting on any part of the body will produce 

 the same progressive motion as if it were immediately ap- 

 plied to the centre of gravity itself (322, 331) and the 

 same rotatory motion as if the centre of gravity were 

 I fixed. [Thus if we imagined the force to be communi- 

 " cated by a particle moving with a given velocity, and at- 

 taching itself to the substance, it is evident, from the pro- 

 perties of the centre of gravity, that the velocity of this 

 point will be the same, whatever be the part of the body 

 to which the particle attaches itself; and, with respect to 

 the velocity of rotation round the centre, it is obvious that 

 this velocity would not be affected by the subsequent appli- 

 cation of any force to the centre of gravity capable of de- 

 stroying the progressive motion, neither will it be affected 

 by the interference of the obstacle, either immediately after, 

 or at, the very beginning of the motion.] The sum of the 

 areas described round the centre of gravity, by the projec- 

 tions of the revolving radii of the different particles on a 

 fixed plane, multiplied by their masses, will always be pro- 

 portional to the rotatory power of the primitive force, pro- 

 jected on the same plane ; and the plane, with respect to 

 which the projection of the momentum is greatest, must 



