IN NON-EUCLIDEAN SPACE. BY A. MCAULAY. 15 



position quaternion due to a clianwing' twist. Similarly 

 below, for hyperbolic space, we have the interpretations 

 of ^'0 = 1 and (p' = —(j>. 



The real linear transformation in hyperbolic space 

 requires that we translate to Joly's notation and then 

 if we please back to our own. Let C be the complex 

 self-conjugate quaternion linity C = S + IV. [Ele- 

 mentary properties of C. C is a square root of K, that 

 is C* = K ; C^ = K2 = I ; C3 = C-i = KG ; C'= C 

 ^ C^ ; C~^ = S — IV. Since C — 1 annuls scalars, 

 and C — I annuls vectors, C satisfies the quadratic 

 (C — l)(C— I) = 0. If 



rj = Cp, (/ = Cp 



then ^.qKr/ = S.j'p, rjKq = S./r. 



Thus K-conjugacy in the system rj, q (our system) 

 corresponds to conjugacy in the system p, y (Joly's 

 system)]. If j) is one of Joly's point or plane symbols 

 (according to Joly, interpreted in Euclidean space) then 

 (/ = C/* is our corresponding hyperbolic position quater- 

 nion. Joly's plane S.y^^o = ^ becomes our plane 

 S.(/K^o = ; Joly's becomes our C(^C~^ ^ i/; ; so that 

 Joly's ^' becomes our ;//\ For real linear transforma- 

 tions, it is Joly's ^, not our ^, which is a real linity. 



From the above standard forms when «■/)'= ih<^ we 

 may derive standard forms for <^' = ±0 by noting the 

 following statements ; if </>' = ±0, then (K^)' = ±K0 

 and conversely ; and also, if ^^ = ±0, then ((/)K)' 

 = ±^K and conversely. I do not see how, similarly, 

 to obtain a standard form for when ^'^ = I ; but such 

 a form may be obtained by translating from our notation 

 for a finite twist t^jq = pq2^'~^t into Joly's notation. 

 First effect a conical rotation about a line through the 

 origin ; then effect a translation along a line through the 

 origin. The first converts, in Joly's notation, q to rqi—'^ ; 

 the second converts q to — K^ + '2pS.pl\.qlS.p^ so that 

 when (p'cj) = 1 we have 



<j>q = — r.l'iq.7'~^ + 2p'S.prlvq7-~^IS.p^. 



This is very different from Joly's standard indeterminate 

 iorm for this case. 



