gb KANSAS UNIVERSITY QUARTERLV. 



k' (k— I) k 

 X= — and /x=rA-a=^ ; — ^ 



k (k— n k' 



k— I k'^i-i k' 



therefore = ; by means of the relation A— — we get after 



k'_i ka-i k 



reduction 



Aa-i— I A^— A 



k= , and k'= . (4) 



Aa — I A^— 1 



When the fixed constant a is given, the conies K and K' corres- 

 ponding to a given value of A are at once determined, 



We can now determine the positions of the conies K and K' for 

 particular values of A. When A=-i, the transformation is an iden- 

 tical one for the whole plane; substituting this value of A in the 

 last equations and evaluating the indeterminate expressions we find 

 k=k'^^. Thus in the case of the identical transformation of the 

 group the conies K and K' are coincident, and touch the line 1 at 

 the point L, such that (A jB,CjL)='^" . When the conies K and 

 K' are coincident, it is easy to see from the construction of the 

 transformation that ever}' point of the plane is unaltered in posi- 

 tion; in other words the transformation in the w-hole plane is an 

 identical one. 



If we consider the construction of an}' transformation T (KK') 

 by means of the conies K and K', we see that the transformation 

 determined by the same two conies taken in the reverse order, 

 T (K'K), is the inverse of the first: i. e. if T (KK') transforms P 

 to P', then T (K'K ) transforms P' back to P; and so with every 

 point of the plane. It is clear that every transformation of the 

 group has an inverse belonging to the same group and that any 

 transformation and its inverse are together equivalent to the iden- 

 tical transformation of the group. 



In considering the positions of the conies which produce a 

 pseudo-transformation of the group it is necessary to consider the 

 value of the constant a. We shall consider the case where a is 

 between o and i, and a real quantity. The coincident conies pro- 

 ducing the identical transformation of the group touch the line 1 

 between Aj and Bj. Let A' gradually decrease in value; then the 

 two conies separate, the point of contact of K approaching Aj and 

 the point of contact of K' approaching B,. When A— o, k=^ — x 

 and k'=o; the conic K then becomes the degenerate conic AAj, 

 while K' becomes BBj, Thus the pseudo-transformation is pro- 



