122 



KANSAS UNIVERSITY SCIENCE BULLETIN. 



a pair of corresponding points from the invariant point is constant for 

 all pairs of corresponding points. Let x be the point at infinity on 

 the line ; t is thus seen to be the reciprocal of the segment Axi where 

 xi is the point into which the point at infinity is transformed. 



Theorem 7. In a transformation of type I, k, the cross-ratio of the 

 invariant points and a pair of corresponding points, is constant for 

 all pairs of corresponding points ; in a transformation of type II, t, 

 the difference of the reciprocals of the distances of a pair of cor- 

 responding points, is constant for all pairs of corresponding points. 



The natural parameters. — When the transformation is written in 

 the form of equation ( 1 ), we see that there are three independent para- 

 meters viz., -|-> -g-» -j-> when it is of type I ; in the case of a 



transformation of type II, the relation, (a + d) 2 = 4(ad — be), is sat- 

 isfied, and there are but two independent parameters. The coeffi- 

 cients, a, b, c, d, have no simple geometric meanings; but in the 

 normal forms A, A', k and A,t have definite important geometric mean- 

 ings. The parameters A, A', k and A, t are called the natural para- 

 meters of the transformation. 



Explicit normal forms. — Equations (15) and (18) may be put 

 into the forms : 



xi = 



, and xi = 



(22) 



These are called the explicit normal forms of types I and II, re- 

 spectively. 



Type lias the limiting form of type I. — It is evident that type 

 II is the limiting form of type I when the two invariant points coin- 

 cide. From equation (16) we see that k=l when A=A'. The 



fraction 



becomes indeterminate when A = A'. Putting for 



A, A' and k their values from ( 16 ), we have : 



lim 



Zc 



A' = A A' — A a+d 



2C i.l i lim k — 1 



But from ( 19 ) ^ = t ; hence, t = J^ A 



(23) 



A A'— A 



By means of this relation the normal form of type II can be de- 

 duced directly from that of type I. Dividing both numerator and 

 denominator of ( 15 ) by A — A', we get : 



Xi = 



(A-kAM 



(A-A') ' 



AA'(1 — k) 



(A-A') 



( 1-k) . kA- A' 



,— AM ■ 2L ~' (X-, 



(A-A') 



(A-A') 



