IN NON-EUCLIDEAN SPACE. BY A. MC AULAY. 5 



Indeed these two line theories in hyperbolic and 

 elliptic spaces are in a sense identical. Our formulae 

 for hyperbolic space become formulae for elliptic space 

 by a simple device. Let P = — 1 as in § 1 above ; let 

 E be a unit for \vhicli E^ = 1 and such that E is 

 commutative with quaternions. E is in fact Clifford's 

 (1). Let J be put for either I or E ; it being understood 

 that when J = I our formulae have real interpretations 

 in hyperbolic space, and when J = E in elliptic space. 

 The reader is advised to ignore the second possibility 

 (J = E) for the present, confining himself to J = I 

 and hyperbolic space. After viewing the hyperbolic 

 development let him return and observe how we might 

 have put J -= E from the beginning. The reason that 

 the notation sinh c, cosh c, etc., is avoided below, and 

 the equivalent notation J~i sin tic, cos tic, etc., is used 

 instead, is that thereby the formulae are left ready for 

 interpretation with the meaning of J , J = E. 



[Note to assist the reader in making the interpretation 

 J = E. li q z= p + Ep' where p and p' are real 

 quaternions so that 5' is a Clifford bi-qualernion, we must 

 first, in fairly oljvious ways, define, V^, Sq, Ky, Tq, Uq, 

 Aq '■= cos~i SVq . Putting Aq = a + Ea' where a 

 and a' are ordinary real scalars, called angle and 

 advance ; there is more difficulty than with complex 

 quaternions in a precise and unambiguous use of these 

 terms. SU^' being = b + E6' we have cos a cos a = b, 

 sin a sin a = —b'; and both b and b' may be positive 

 or negative. Remembering that in elliptic space a twist 

 with a and a for advance and angle, about a given axis, 

 is the same as a twist with a and a for advance and 

 angle, about the polar axis ; we see that any rule for 

 determining angle a and advance a from b and b' ought 

 to be ambiguous to the extent that the two are inter- 

 changeable. In view of all this I prefer the following 

 rule : — Let one of the two a and a be between and tt, 

 and the other between ^tt and — ^tt ; in the case of a 

 pure translator or versor let the range of values be, 

 0, tt]. 



§4. We shall pass lightly over such parts of the 

 treatment of hyperbolic space as are suggested by §2 

 above. 



