EMCH: INVOLIJTORIC COLLINEATIONS IN PLANE AND IN SPACE. 207 



aa, 

 — aa^ 

 as is also seen from the resulting involution (4). 



Transformation (3) being a perspective collineation in which 

 the center lies in the axis of collineation, b jX ^ -j-c ^y^ =0, and 

 where ^ = ~f-i, is sometimes called an elation.'''' All the elations 

 which leave the center invariant form a two-termed group, so that 

 we now have the result: 



TJic effect of the two-tenned group of elations upon the involutions 

 that leai'e a certain center invariant is the same system of involutions. 



We will investigate how this result is to be modified if we apply 

 the two-termed group of elations to a certain involution with the 

 coefficients a, b, c. Evidently we may put 



a^a^A 

 bja-]-ajb=B 

 Cja-!-ajC = C 

 where A, B, and C are three independent arbitrary real numbers. 

 In fact, we can choose the three coefificients a^, bj, and c,, or an 

 elation out of the group, in such a manner that these equations of 

 condition are satisfied. This is the case if 



■ A • 



b, 



a 



aB— Ab 



'' ~^^ ' 



aC— Ac 



a« 

 Hence: 



The effect of the two-tennecl group of elations upon a certain invo- 

 lution is the system of all involutions, that leave the center invariant. 



The axes of the elation, and the original and the resulting 

 involutions, are represented by 



b,Xi + Cjy^=o, 

 bx-j-cy=2c, 

 (bja4-aob)x-)-(Cja-pa^c)y=2aja, 

 respectively, and intersect each other in the point 



2aCi — 2ab, 



bc^— bjc' -^ bCj— bjc' 

 If, therefore, the axes of the elation and the original involution 

 have fixed positions, this point is invariant, which implies the 

 theorem: 



*Sophus Lie, loc. clt. pa^e 26-. 



