NKWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 121 



which is of the same form as 



ax + b 



Xl "" ex +d' 



Comparing the coefficients of these forms, we have 



A-kA' a A *' b kA-A' d. 



1-k — 7' AJ ^ — c' 1-k ~ c' 



solving for A, A' and k, we find 



A a — d+l/(a+d) 3 — 4(ad — be) 



A = zl ' 



A' a — d — i/u+d) 2 — 4(ad— be) (\P)\ 



i (a + rt + y^a+d)* -4Ud-bc) ) 8 



K 4(ad — be) 



The values of A and A' thus obtained are the same as the roots of 

 equation (5). Equation (13) is called the implicit normal form of 

 type I. 



Normal form of type II. — A transformation of type II, whose 

 single invariant point is A, is reducible to the form 



To verify this, solve for xi ; thus, 



(1 +tA)x- t'A /io\ 



Xl= tx + (l-tA) ' ( lb ) 



This is the same form as (1). A and t are found in terms of a, b, c, 

 d, as before, by comparing coefficients and solving for A and t ; thus, 



A = 5 S S andt = ; ^ H . (19) 



Equation ( 17 ) is called the implicit normal form of type II. 



Theorem 6. Every transformation of the form xi = " + d is re- 

 ducible to one or the other of the normal forms 



x t -A' _i x-A' or 1 — _L__l t 

 x,-A K x-A OX x,-A x-A^ 1 ' 



Geometrical interpretation of the normal forms. — The normal 

 form of type I may be written : 



k=f^r^=( AA ' x *'>; ( 2 0) 



i.e., k is the cross-ratio of the four points A, A', x, xi, where A and 

 A' are the invariant points, and x and xi a pair of corresponding 

 points. Here x and xi are any pair of corresponding points, and k is 

 a constant quantity. 



In the normal form of type II the expressions x — A and xi — A 

 are the distances of a pair of corresponding points from the invariant 

 point. The normal form of type II may be written : 



— l ~T L T=t> ( 21 ) 



X! — Ax — A ' *■ ' 



which shows that the difference of the reciprocals of the distances of 



