134 THE THEORY OF SCREWS. [145, 



for the pairs of related screws becomes still more simplified by the theorem, 



that 



When the system is one of dynamical involution, the chord joining an 

 impulsive screw with its instantaneous screw passes through the pole of 

 the homograpldc axis. 



We may take the opportunity of remarking, that dynamical involution 

 is not confined to the system of the second order. It may be extended to a 

 rigid body with any number of degrees of freedom, or even to any system of 

 rigid bodies. Whenever it happens that the relation of impulsive screw and 

 instantaneous screw is interchangeable in one case, it is interchangeable in 

 every case. 



For, let O l , ... 6 n be the co-ordinates of an instantaneous screw, then ( 97) 

 the corresponding impulsive screw has for co-ordinates, 



MI n u n a 



v ly ... v n ; 



PI Pn 



and if this latter were regarded as an instantaneous screw, then its impulsive 

 screw would be 



3 * 



Pn 



but as this is to be only 

 we must have 



which shows that if the theorem be true for one pair, it is true for all. The 

 conditions, of course, are, that any one of the following systems of equations 

 be satisfied : 



1/2 ,.2 ,,. 2 



, &quot;i _ i III _ i U n 



~ Pi ~ P 2 ~ ~ Pn 



146. Another Construction for the Twist Velocity acquired 

 by an Impulse. 



Reverting to the general case, we find that the chord A A (Fig. 27) is cut 

 by the homographic axis at T, so that the square of the acquired twist 

 velocity is proportional to the ratio of TA to TA . 



For, with the construction in 142, draw HQ parallel to AT; then, 



AT 



