480 THE THEORY OF SCREWS. [432- 



If, however, the vector be left, then Y must be displaced to a distance F , 



defined by 



H log (XX 00 ) = H log (FF P P) : 



we therefore have 



H log ( YY PP ) = Hlog( FF P P) ; 

 but from an obvious property of the logarithms, 



H log ( FF P P) = - H log ( FF PP ) ; 

 whence, finally, 



H log ( YY PP ) = - fl&quot; log ( FF PP ). 



We hence have the important result, that the intervene through which a 

 point on one of the common conjugate polars is displaced by one of two 

 heteronymous vectors of equal amplitude, merely differs in sign from the 

 displacement which the same point would receive from the other vector. 



433. The Conception of Force in non-Euclidian Space. 



In ordinary space we are quite familiar with the perfect identity which 

 subsists between the composition of small rotations and the composition of 

 forces. We shall now learn that what we so well know in ordinary space is 

 but the survival, in an attenuated form, of a much more complete theory 

 in non-Euclidian space. We have in non-Euclidian space force-motors and 

 force-vectors, just as we have displacement-motors and displacement-vectors. 

 We shall base the Dynamical theory on an elementary principle in the 

 theory of Energy. Suppose that a force of intensity / act on a particle 

 which is displaced in a direction directly opposed to the force through a 

 distance 8, then the quantity of work done is denoted by fS. 



434. Neutrality of Heteronymous Vectors. 



We are now able to demonstrate a very important theorem which lies at 

 the foundation of all the applications of Dynamics in non-Euclidian space. 

 The virtual moment of a force-vector and a displacement-vector will always 

 vanish when the vectors are homonymous and at right angles. The analogies 

 of ordinary geometry would have suggested this result, and it is easily shown 

 to be true. If, however, the two vectors be not homonymous, the result is 

 extremely remarkable. The two vectors must then have their virtual moment 

 zero under all circumstances. 



The proof of this singular proposition is very simple. Let the two 

 vectors be what they may, we can always find one pair of conjugate polars 

 which belong to them both. Let the two forces be X, A, on the two conjugate 

 polars, and let the displacements be p, p, then the work done is 



X/4 XyU, = 0. 



