IN NON-EUCLIDEAN SPACE. BY A. MC AULAY. 9 



(8) If u and v are intersecting planes take the 

 origin on their line of intersection. If they are non- 

 intersecting planes take the the origin at the point where 

 their common perpendicular meets one of them. [If 

 they are parallel planes Y .vu—'^ as a matter of fact is a 

 nullitat and S.y?^"^ = 1 ; proof of which is left to 

 reader.] If either u or v is a point take that point for 

 origin. Then the following interpretations are rendered 

 evident. (i) When u and v ai'e intersecting planes vu~'^ 

 is an ordinary versor whose axis is definitely fixed 

 in space (the axis is the line of intersection) and whose 

 arc is the angle between the planes. (ii) When u and v 

 are non-intersecting planes or when v and v are two 

 points vu~^ is a translator whose axis is the common 

 perpendicular of the planes or the line joining the points, 

 and whose advance is the distance between the planes or 

 the points. (iii) When u and v are, the one a point and 

 the other a plane, vu~^ is a nnitat whose angle is \it, 

 whose advance is the distance between the point and 

 plane, and whose axis is the perpendicular from point to 

 plane. It Avill be seen that a unit vector, with definite 

 axis fixed in space, occurs twice among these interpreta- 

 tions. Under (i) it occurs as the ratio of two intersecting 

 perpendicular planes ; under (iii) it occurs as the ratio of 

 a plane and an incident point. With our present 

 geometric interpretations this geometric concept of a 

 directed unit line with axis fixed in space ought not to 

 be called a vector ; for the future we shall call it a unit 

 rotor. 



(9) Turn the unit rotor i througli an angle a 

 towards the rotor J; then translate the turned rotor 

 along the rotor A to a distance a. The first change of 

 position is effected by the o))erator e*"'"^ ( ) g-^"''^ and the 

 second by the operator e*"^"'''' ( ) e"^"^"'''^ ; that is to say i 

 is first changed to e'^^H and then to e(" + "^"'''''i. Since i 

 and the final ^f^ + JfO^i-i are any two unit rotors, we have 

 here proved that the ratio e'e"^ of any unit rotor z to any 

 othpr £ is the unitat whose axis is the common perpen- 

 dicular between e and e', and whose angle and advance 

 are the angle and distance of the twist about this 

 eommon perpendicidar, which converts the one into the 

 other. We have geometrically interpreted the general 

 ■complex unitat and incidentally justified our terms 

 angle, advance, versor, translator. 



