IK NON-EUCLIDEAN SPACE. BY A. MC AULAY. ^ 



(4) In hyperbolic space, contrary to the case in 

 elliptic space, we cannot have S{um~1) = for two real 

 points. To obtain the equation of a plane in the form 

 SjoK^' = we are led to define the position nnitat of the 

 plane throuofh P (position imitat u) perpendicular to 

 OP, as uUYu. P being given ; and Q, with position 

 unitat y, being variable ; Q, is on the plane provided 



SyK(MUVM) = 0. 



This is easily proved from (6) below by taking the 

 origin at P, so that the position unitats of Q and O 

 become vu~'^ and w"i. Expressing that cos QPO = 

 we get the the equation just written. The characteristic 

 of such a position unitat of a plane is that the angle has 

 the definite value ^tt ; for a point the definite value was 

 zero. Taking v. = exp (eJcj we have in full 



ulJYu = 6 exp (eJc) = - sin Jc + a cos Jc 



= exp (£[^7r rf Jc]) 



from which we see that the scalar is a negative pure 

 imaginary and the vector is real. The plane passes 

 through the origin when c is zero. In this case its 

 position unitat is the real vector e. 



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

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



xu + yv + zw = Q 



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

 satisfy the sine formula 



a-i sin (J. QR) = 3/-1 sin (J. RP ) = z-^ sin (J. PQ). 



§6. We have now to make some similar fundamental 

 statements which are confined in application to the 

 present method, being inapplicable or unnecessary with 

 the method of §2. 



(6) There is no special mathematical property dis- 

 tinguishing the origin O from any other origin P. 

 At first sight this statement seems inconsistent with 

 the prescription that vector parts of line-segments 

 through O are pure imaginaries, whereas with other 

 origins this is not so. But this is a mere question 

 of terms, not one of contained mathematical meaning. 

 If we translate i, /, k, three real rectangular unit vectors 



