296] DEVELOPMENTS OF THE DYNAMICAL THEORY. 315 



of the homographic equation is based on the fact that if X, \ be two 

 instantaneous screws and p, p the corresponding impulsive screws, then 

 the formula 



COS (X + P * t , x COS (\p f ) = 2tsr AX 



, , t , x 



cos (X/o) cos \Kff) 



must be satisfied. 



And generally it is satisfied. In the case of two cylindroids with normal 

 planes it is however easy to show that there are certain pairs of screws for 

 which this formula cannot obtain. 



For in such a case there is one screw X on A which is perpendicular to 

 every screw on P, so that whatever be the p corresponding to X, 



cos (\p) 0. 



Since no other screw X can be perpendicular to any screw on P we cannot 

 have either 



cos (X p), or cos (A///), zero. 



Hence this equation cannot be satisfied and the argument that the homo- 

 graphic equation defines corresponding pairs is in this case invalid. 



We might have explained the matter in the following manner. 



When the principal planes of A and P are normal there is one screw X 

 on A which is perpendicular to all the screws on P. If therefore the two 

 cylindroids were to be impulsive and instantaneous, there must be a screw 

 on P which corresponds to X. It can be shown in general ( 301) that 



d\ = p\ tan (X0) 



when dx is the perpendicular from the centre of gravity on X; it follows 

 that when (A.0) = 90 we must have either p^ zero or d\ infinite. 



But of course it will not generally be the case that X. happens to be one 

 of the screws of zero pitch on A. Hence we are reduced to the other 

 alternative 



d*. = infinity. 



This means that the centre of gravity is to be at infinity. 



But when the centre of gravity of the body is at infinity a remarkable 

 consequence follows. All the instantaneous screws must be parallel. 



For if 6 be the impulsive wrench corresponding to X as the instantaneous 

 screw, then we know that 



di=p*. tan(X0), 

 and that the centre of gravity lies in a right line parallel to X and distant 





