b QUATERNIONS APPLIED TO PHYSICS 



Let a translator 



u = cos Jc + £ sin Jc = exp (eJc) 



where c is a real scalar and e a real unit vector, represent 

 a line-segment through a given point O of hyperbolic 

 space ; and if OP is in the direction e and of length c 

 let u be taken as the position unitat of P (origin O). 

 If with origin O, n, v are the position unitats of points 

 P, Q vu-^ signifies the line-segment PQ. Expandji?M-i 

 in full thus ; if 



u = cos Jc + £ sin Jc, and v = cos Jc' + e sin Jc' 



then 



rw~i = vKu 



= (cos Jc cos Jc' - sin Jc sin Jc' Ste') 



+ (— £ sin Jc cos Jc' + £' sin Jc' cos Jc 



+ \te sin Jc sin Jc'j 

 = cos Jc" -f- a" sin Jc" 



where c" is a real scalar and c" is a unit vector (the state- 

 ment meaning that f"^ = — 1 ), which in general is 

 complex. It is important to note that the necessary and 

 sufficient condition for e" to be real is 



Yee. sin Jc. sin Jc' = 

 or P, Q and the origin are collinear. Since t" is in 

 general complex vit'^ does not belong to the triply 

 infinite system of u, v. On the other hand vii'^ is not a 

 general unitat since S.vu~^ is real. It is in fact the 

 general form of a translator ; satisfying the single 

 scalar relation that its angle is zero. 



§5. This jtroperty that ivr"i belonors to a quintuple 

 infinite system and not to the triply infinite system 

 of u, V is of course an important distinction from the 

 case for elliptic space of §2 above. Nevertheless we 

 have five fundamental statements similar to those 

 numbered above for elliptic space. 



(1) J-iA(r?/-i) equals distance PQ ; or, what is 

 the same, Sivw''-) = cos (J.PQ). 



(2) If IV is the position unitat, of a third point R j 

 wv~'^ signifies the same line-segment QR, whether O or 

 any other point P be taken as origin. 



(3) nvu-^ is the line-segment v translated along the 

 axis of u to a distance 2J~iA?/. 



