264 PROFESSOK TAIT ON THE ROTATION OF A 



§§ 1-14. Kinematics of a Bigid System with one Point fixed. 



1. If g represent the instantaneous axis of a rigid body, its length being 

 employed to denote the angular velocity about it ; then, <& being the vector of any 

 point of the body, drawn to a point in the axis as origin, we obviously have 

 (using Newton's convenient notation) 



*=w- Y - «■ 



This formula was given long ago by Hamilton. 



2. Every infinitely small displacement of a Rigid System, one point of wh ich 

 is fixed, takes place about an instantaneous axis. 



Let a-, ©■„ be the vectors of any two points of the system, referred to the fixed 

 point as origin ; then, whatever displacements may occur, we must have (on 

 account of the rigidity of the system) 



Ta- = const. , TV, = const. , Sarar, = const. 

 Hence, differentiating with respect to t, 



Sarar = , Sar-jar, = , Sarar, + Sarar j = ... (2). 



The first shows that 



ar = Vsar , 



where g is some vector. With this the third gives 



S . ar ( \ ear, — ar.,) = , 



which must be true for all values of ar. Hence we have also 



ar, = V Ear, . 



This is consistent with the second of equations (2), so that the existence of the 

 instantaneous axis is proved. From the fact of its existence follows at once the 

 representation of the motion, in every case, by the rolling of a cone fixed in the 

 rigid system upon another cone fixed in space. The case of finite displacements 

 will be treated farther on (§ 5 below). 



3. To find the instantaneous axis, when the rectors, and vector-velocities, of 

 any two points of the system are given. 



Here we have to find g from the two equations 



ar = V ear , ar = V Sai^ . 



They give by inspection 



V arra", = — soarar, = SOarar. , 



or, more symmetrically, 



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