NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 123 



Putting A'= A and A , H jf A ^j-, = t, this reduces to ( 18 ). ■ 



In the explicit normal form of type I, (22), subtract the second 



row from the last, divide through by A' — A, and pass to the limit. 



In this way we get the explicit normal form of type II. 



Resultant of T and 71. — We next consider the resultant of two 



transformations T and Ti, both of type I, which have no invariant 



point in common, and which are given in their explicit normal forms. 



T and Ti are as follows : 



_ (A-kA')x-AA'(l-k) i (A.-k.A',)*, -A,A',(l-k,) 



Xl_ (l-k)x-(A'-kA) clI1U * 2 (l-k 1 )x 1 -(A' 1 -k l A 1 ) 



Eliminating xi we have : 



j(A-kA') (Aj-k.A'J-AA'd-k) (l-k,)| x 



X2 = ~, T~ 



| (1-k) (A-kA')-d-k,) (A'i-k^,) ^X 



j(AA'd-k) (A.-k.A'J-A.A'.d-k.) (\'-V\')\ 



] AA'd-k) d-k,)-(A'-kA) (A'x-k.A.I j- 



px — q (A, — k ? A'o)x — A 2 A' 2 d — k 2 ) 



rx — a ~ (1 — k 2 )x— (A' 2 — k 2 A Q ) 



We readily find : 



A 2 A' 2 = 4' A 2 + A' 2 = -^> A 2 — A' 2 =^^ 



(24) 



1 +k 2 



V ( P + s I ■ — 4 qr 

 2 iV 2 : 



A.— AV — " /(p + i,B,T - qF 



Hence, 



^g(p-s)=V(p-hs) 2 -4qr. 

 From this we find : 



1 + k, p-s _d-kk,) (A -A',) (A'-A,) + (k +k,) (A -A,) (A' -A',) 



•j/ka i/qr-pa V kk t (A — A') (A 1 -A' 1 ) 



1-k -k^+kk n , JL±k i> (25) 



l.l. i ■./ 1,1, v / 



v'kki v' kk t 



where 



one of the cross-ratios of the four points A, A'i, Ai, A'. 

 We also find 



If one or both of the transformations T, Ti are of type II, the 

 resultant is obtained by putting A'=A and lim A -£ L A 7 = t, or A'i = A' 

 and lim^z k ^_ = t 1 ,orboth,asthecasemaybe,in (24), (25), and (26.) 



§ 3. One-parameter Groups of Projective Transformations. 



Resultant of T and 71 with common invariant points. — Let T 

 and Ti be two transformations of type I having the same invariant 

 points A and A', and let T transform the point x to xi, and let Ti 

 transform xi to X2. The resultant of T and Ti also leaves A and A' in- 



