IN NON-EUCLIDEAN SPACE. BY A. MCAULAY. 1^ 



rotor-couples, each of magnitude ^a. Then combine 

 one of these with ^a at A and the other Avith the 

 remaining ^a at A. The effect is easily deduced from 

 the above and may be stated as follows. The rotor at 

 A is equivalent to a component rotor at C, the magnitude 

 of the component being a multiplied by cos [J. distance 

 CA] ; together with a moment, that is a rotor-couple 

 through C, the magnitude of the moment being a multi- 

 plied by J~^ sin [J. distance CA]. 



§8. Let w -h J(T = (t>' -I- Jcr' where w, <y are rotors 

 through O, and w', ct' rotors through P, and let the 

 position unitat of P (origin O) be u. By §7 the com- 

 ponent and moment of a>, at P, are ^(w + i/ijju-^) and 

 ^{w — uii)U~^) respectively ; and the component and 

 moment of Jo-, at P are |J(o- + uau~^) and hH<T — uaii-^) 

 respectively. Hence 



(1> 



: E. 



In the notation of §2 where E is not used, the same 

 reasoning applies ; rotor w at O gives rotor ^(w + vww'^) 

 and rotor-couple i(w — Uh)U—^) at P, and rotor-couple a 

 at O gives rotor-couple h{(r - naii-^) and couple of rotor- 

 couples (i.e. rotor) ^{a — iiaw^) at P ; or 



u}' = Mo> + a) + i'/(w — o-)«~' \ 

 a = h{w + (t) + h>fi — 10 + (t)m"' j 



W + (t' = (jO + (T \ /.2\ 



h) — a ^ uyM — o-)/<^ / ' 



In addition to (2) we have the statements from §2 above 

 that left and right parallels at P, of w at O, are denoted 

 by a> and utou "^ ; and similarly for a. Hence component 

 and moment, [at P], of w at O, are ^(left -f- right 

 parallels) of w at O, and ^(left — right parallels) of hy 

 at O, respectively ; and similarly for a. 



Real Linear Transformations of Points and Planes. 



§9. The explanation of the meaning of (1) of §8 is 

 more concise than that of (2) ; and when motors naturally 



