245-] PLANE MOTION. 133 



(6) A point P describes an ellipse owing to a central acceleration 

 f(f) =i*>/r 2 directed toward the focus S. Its initial velocity V Q makes an 

 angle i// with the initial radius vector r . Determine the semi-axes a, b 

 of the ellipse in magnitude and position. 



244. The student will find numerous examples for further practice 

 in the kinematics of a particle in the following works : P. G. TAIT and 

 W. J. STEELE, A treatise on dynamics of a particle, 6th ed., London, 

 Macmillan, 1889 ; W. H. BESANT, A treatise on dynamics, London, Bell, 

 1885 ; B. WILLIAMSON and F. A. TARLETON, An elementary treatise on 

 dynamics, 2d ed., London, Longmans, 1889; W. WALTON, Collection oj 

 problems in illustration of the principles of theoretical mechanics, 3d ed., 

 Cambridge, Deighton, 1876. 



5. VELOCITIES IN THE RIGID BODY. 



245. A rigid body is 1 said to have plane motion when all its 

 points move in parallel planes. Its motion is then fully deter- 

 mined by the motion of any plane section of the body in its 

 plane. 



It has been shown in Arts. 18-24 that the continuous motion 

 of an invariable plane figure in its plane consists in a series of 

 infinitesimal rotations about the successive instantaneous cen- 

 tres, i.e. about the points of the space centrode. 



If at any instant the centre of rotation and the angular veloc- 

 ity to about it be known, we can find the velocity of any point of 

 the plane figure. 



To show this let us first take the instantaneous centre as 

 origin. Then the component velocities v x , v y of any point P 

 whose co-ordinates are x, y, or r, 6, are found (Art. 141) by dif- 

 ferentiating the expressions 



with respect to /. Considering that dQ/dt is the angular 

 velocity o> about the instantaneous centre, we find 



