428] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 475 



We thus learn the remarkable fact, that if a right (left) vector be followed 

 by a left (right) vector, the effect produced is the same as if the order of 

 the two vectors had been interchanged. 



This is not true for two right vectors or two left vectors. 



The theorems at which we have arrived may be thus generally 

 enunciated : 



In the composition of vectors the order of two heteronymous vectors does 

 not affect the result, but that of two homonymous vectors does affect the result. 



In the composition of two homonymous vectors the result is also an homony 

 mous vector. In the composition of two heteronymous vectors the result is not 

 a vector at all. 



The theorems just established constitute the first of the fundamental 

 principles relating to the Theory of Screws in non-Euclidian Space referred 

 to in 396. Their importance is such that it may be desirable to give a 

 geometrical investigation. 



428. Geometrical proof that two Homonymous Vectors com 

 pound into one Homonymous Vector. 



Left vectors cannot disturb any 

 right generators of the infinite quad- 

 ric. Take two such generators, AB 

 and A H (Fig. 48). Let AA , 

 BB , CC be three left generators 

 which the first vector conveys to 

 A^Ai, -BjjB/, eft-!, and the second 

 vector further conveys to A 2 A 2 , 

 B,B 2 , C 2 C 2 . Let X and Y be the 

 double points of the two homo- 

 graphic systems defined by A, B, G 

 and A 2 , B 2 , (7 2 . Then we have 



and 



As anharmonic ratios cannot be 

 altered by any rigid displacement, it 

 follows that X and Y must each 

 occupy the same position after the 

 second vector which they had before 

 the first, similarly, X and Y will 

 remain unchanged, and as the two 

 rays, AB and A B are divided homo- lg 8 



