4 QUATERNIONS APPLIED TO PHYSICS 



line-segments, one through O, the other through C, each 

 of which is represented by qrq~'^, are left parallels in 

 Clifford's sense. This shows the connection between the 

 (doubly infinite) several line-segment meanings we have 

 assigned to a quaternion p. They have equal tensors 

 and arcs and their axes are any left parallels. rq~'^ 

 represents AB in the diagram ; therefore it represents 

 the left parallel of AB through O ; therefore the 

 right parallel of AB through O (obtained from the 

 left parallel by conical rotation round OA) must be 

 ^~i irq~'^)q =^ q~^r. Thus the two parallels through O. 

 left and right, of the line-segment from the point q 

 (origin O) to the point r, are rq~^ and q~^r respectively. 



(4) If Q (position unitat, r), a given point, and P 

 (position unitat, ?<) a variable point, are such that 

 S{vu-^) ^ 0, P is quadrantally distant from Q, that is 

 to say the locus of P is the polar plane of Q. Hence 

 S.pKq = (/> given, q current) is the equation of the 

 plane polar toyj. 



(5) If u, r, w are the position unitats of three 

 points P, Q, R ; if x, //, z are scalars ; and if 



xu + 7/v -t zw = ; 

 then P, Q, K are collinear, and their mutual distances 

 satisfy the sine formula 



a-i sin Qll = ,?/-' sin RP = z-^ sin PQ. 

 For a: + yvii~^ + ztvu~^ =0, so that t/Vwm"! + zYivw^ 

 = 0. 



Clearly (4) and (5) show that one application of 

 the present method is very similar to Joly's use of q as 

 a point or plane symbol in Euclidean space (Joly's 

 Manual of Quaternions^ chap. xvii). As will be briefly 

 explained below, Joly's method is much better adapted 

 to interpretations in hyperbolic space than in Euclidean 

 or elliptic. His (real) unit sphere has to be interpreted 

 as the (real) absolute of hyperbolic space. 



Hy])erbolic Space Preliininaries. 



§3. In hyperbolic space, our initial principles, as 

 appearing in §2, of mathematical necessity, carry us on 

 to a line calculus, a calculus precisely of the type of 

 Clifford's bi-quaternion calculus for elliptic space. 



