Manchester Memoirs, Vol. Hi. (1908), No. 8. 3 



positions. We shall first obtain the magnitudes and 

 directions of the impulses necessary to impose the motion. 

 Let P be the instantaneous centre of rotation of a 

 body of mass J/, and with mass centre C. If an impulse, 

 /, acts at P in a direction making a small angle, 0, with 

 PC^ it gives the mass centre a velocity //i/, and at the 

 same time produces an angular velocity IJi<^\Mk'^% where 

 h = PC d^n^ k is the radius of gyration about C. 



Fig. I. 

 The velocity of P after the impulse is thus made 

 up of the components PQ = I/M along the line of the 

 impulse, and QR = (Ik'^/Mk'^)^ perpendicular to PC: the 

 resultant is therefore PR. In our problem this velocity 

 is constant (numerically), = V. PQR being approxi- 

 mately a right angle, PQ differs from PR by a small 

 quantity of the second order : hence the impulse 

 I=M.PQ = MV so far as quantities of the first order are 

 concerned. Also if RPC is denoted by d 



d_ SR ;r + /^" 



Thus to impress the required velocity F on P in a 

 direction at an angle Q to the rod an impulse J/ F must 



act at an angle ^,—^9 to the rod. An impulse of double 



magnitude will then reverse the motion of P from 

 Fto— V : and the imposed vibratory motion of P with 

 constant speed V in the path AB is produced by impulses 



