EiMCH: INVOI.UIORIC COLI.INKAI'IONS IN PLANE AND IN SPACE. 215 



with the invariant points ± i ■% _^ , where k designates a para- 



^ c 



meter and b and c the same constants as in (6), leaves the 



character of the invokition (7) unchanged. The transformation 



(8) applied to (7) gives 



(ab— bck)u + (b2+abk) 



u.. 



(ack-}-bc)u — (ab — bck) 



(9) 



Assuming — as the factor of proportionality, we obtain as the 



resulting involution in space 



(ab— bck)x+(b2+abk)y 



y-2-- 



Zc 



(ack-pbc)x — (ab — bck)y 



(10) 



(ab— bck)2 4-(ack + bc)(b2+abk) 



The transformation in space which produces this result is 



Xg^bxj — bkyj, 



yo=kcXi+byj, (II) 



Z3==Zj(b3+bck2). 

 These equations are obtained from (8) and (g) by noticing that the 

 determinant of the transformation (g) is 



(a2+bc) (b3+bck3). 

 The transformations as represented by equations (11) is linear and 

 the determinant is 



b — bk o 



kc b o =(b2-f bck)-',^ 



o o b»+bck--^ 



i. e., always positive. 

 The invariant rays are 



y \ c ' 



and z=o, or 00 . 



These lines are real or imaginary according as b and c are of 

 apposite or same sign. Thus, the system of linear transformations 

 (6) leaves a tetrahedron invariant which has one of its sides at 

 infinity, and as its finite edges 



x=y=o; 



b X 



\ c y 



\ c 



