IN NON-EUCLIDEAN SPACE. BY A. MCAULAY. 11 



Straight Lines Symmetric About a Point; Components 

 and Monients ; Rotors and Rotor-couples* 



§7. In elliptic space there is a special method of 

 resolution (by left and ri^ht parallels) of forces and the 

 like which is inapplicatjle to hyperbolic space ; but 

 resolutions in the latter easily translate to corresponding 

 resolutions in the former. Let us, then, pay special 

 attention to hyperbolic resolutions. The terms " equal 

 and opposite " and " equal and similarly directed," as 

 applied to rotors, may be based on the conception of the 

 straight line symmetric, 

 about a given point, to a 

 given straight line. Let 

 AB be a given straight line 

 and C a given point. Join 

 the points A, B, .... to C 

 and produce to A', B', . . . . , 

 making AC equal to CA' 



etc. The locus of A', B', .... is the straight line 

 symmetric to AB, about C. C is the centre of symmetry, 

 and the straight line through C perpendicular to the 

 plane CAB is the axis of symmetry ; AB and B'A' are 

 similarly directed ; AB and A'B' are opposite. [In 

 hyperbolic space, AB and B'A' are any two non-inter- 

 secting coplanar straight lines ; in elliptic space they are 

 any two coplanar straight lines whatever. In hyperbolia 

 space there is only one centre and one axis of symmetry. 

 In elliptic space there are two axes, and for polar space 

 two centres, for antipodal space two pairs of antipodal 

 centres. Tlie terms "similarly directed" and 'oppo- 

 site " give different meanings for the two axes ; and 

 except when the centre or axis of symmetry is given the 

 terms ought not to be used in elliptic space ; unless we 

 elect to imply that the " near " centre and axis is to be 

 understood, to the exclusion of the " far " centre and 

 axis.] The centre of symmetry is the mid-point of the 

 common perpendicular MM' of the two straight lines. 

 Below the length CM is taken to be z ; the unit rotor 

 along CM is taken to be ^ ; and the unit rotor through 

 C, perpendicular to A, in the plane CAB, is taken to be 

 ;'. For application in §8 below, it is important to 



* Probably the contents of §7 are well known. If that is so, no liarrn cart 

 come from again enunciating what happens to be important for our purposes- 



