12 QUATERNIONS APPLIED TO PHYSICS 



remark, that the one straight line B'A' may be obtained 

 from the other AB by a translation 2AC through any 

 point A of the second mentioned. 



Equal and similarly directed unit rotors along AB 

 and B'A' are 



g-JzAy ^^ j ^>Qg J 2- ^ I gjj^ ^2. 



Equal and opposite unit rotors are obtained by reversing 

 the first of these. Thus two equal and similarly directed 

 unit rotors combine to a similarly directed rotor through 

 the point of symmetry, whose magnitude is 2 cos Jz ; 

 and two equal and opposite unit rotors combine to a 

 rotoi'-couple (2z sin Jz) along the axis of symmetry 

 whose magnitude is 2j~i sin Jr and whose sense is that 

 which we should expect from Euclidean analogies. 

 Call this sense the usual sense, and call the opposite 

 sense ths unusual sense. If we combine tw^o equal and 

 opposite rotor-couples by changing the above / to JJ we 

 get 2Je sin Jr. In hyperl)olic space this gives us an 

 unexpected result ; namely, that a couple of rotor- 

 <?ouples has the unusual sense. The anomaly does not 

 occur in elliptic space. 



The anomaly is sufficiently important to deserve a 

 kinematical comment. If we combine a right-handed 

 velocity of rotation a about AB, in the diagram, with 

 an equal one about A'B' we get as we should expect a 

 velocity of translation along the axis of symmetry, in 

 the usual sense, equal to 2a sinh z. On the other hand, 

 if we combine a velocity of translation a along AB with 

 an equal one along A'B' we get in hyperbolic space, as 

 we should scarcely expect, a velocity of rotation 

 {2«' sinh z) in the direction of the curved arrow in the 

 diagram. [This is neither oversight nor nonsense. In 

 Euclidean space, in a similar case the result would be 

 zero. Let tlie reader ask himself Avhat he means by the 

 combination of two velocities of translation of a rigid 

 body.] 



A rotor or rotor-couple is given at a given point A, 

 the magnitude being a. It is required to replace it by an 

 equivalent rotor and rotor-couple at a second given point 

 C. Join AC and produce AC to A' making AC = CA'; 

 and at A' introduce two equal and opposite rotors or 



