TYPES AND NORMAL FORMS. 19 
T-‘after each side of this equation and we get TS’T7*=S, 
which shows the relation between S and S’. 
25. Equations of Sand S’. Let S be given by the equation 
— ae+h bi ax +b 
ies qaet+d cx + d° 
Since S’ is given by S’= T-1ST, we find the equation for S’, 
and let us operate on S by 7, given by «,= 
(—ada—bdeitachi+bed:) «+ (abai+b2c:—a%bi—abdi) 
( —eda,—d?e1+ 2b + edd; )x + (beai+bdei—acbhi—add; ) 
Ls ay! «+ bi! 
~ efa+d)" 
= 
(28 ) 
Let the natural parameters of S be k, A, and A’; and of S’, 
k,, A,, and A,’; we wish to find the relation existing between 
the natural parameters of S and S’. 
From the equations of S’ and S we readily find 
a, +d/= —A(a,+d,), (29) 
where A is the determinant of 7. Also from theorem 6 we 
ay’ bi’ | _|a hh 
have, Ge leeAah a (30) 
The value of k,, the cross-ratio of S’, is by (15) 
= (aa!+ di’ — V(ai'+ di’) 2—4 (ar! dy’ — bier) )? 
ae 4 (ay di’ —by cr) . 
Substituting from (29) and (30) we get 
k, ie (ai+ di — V(a+di)?—4(aidi— bici) )” ewe 
4 (a ai— bi ci) 
If the value of A, one of the invariant points of S, be sub- 
stituted for «in the equation, «,= a x, will be found to 
be equal to A,, an invariant point of S’.. Thus T transforms 
the invariant points of S into those of S’. 
THEOREM 11. When 7 operates on S to produce S’, according 
to the formula S’/= 7-7 ST, the invariant points of S are transformed 
by T into the invariant points of S’ and the cross-ratio of S’ is the 
same as that of S. 
