ONE-DIMENSIONAL PROJECTIVE TRANSFORMATIONS. 7 
this function; we shall designate the cross-ratio by the sym- 
poles = (arse at): 
Let the four points a, 2’, w’, x”, be transformed into 
v,, &,', &;'", x,/’ respectively by the projective transformation 
) aneb 
aoe cet, 
We wish to compare the cross-ratio of these four points with 
that of their four corresponding points. To do this we have 
only to substitute (1) in the cross-ratio function and reduce 
the resulting expression. Thus 
ax’ +b ax+b ax/’+b ax+b 
gi! — 0 , a! —m ca! +d ce+d _cx!"+d cx +d 
a! — any! al—ay a+b ae’ +b (ael/+b  ax'+b 
ca! +d " cx’ +d cal’td sf cx'+d 
(ad— be) (cx/+d) (w"—ax) , (ad—be) (ex’+d) (x"'/—2) 
~ (ad—be) (ex+d) (x"—a!') ~ (ad—be) (ex+d) (%"’—2x’) 
a 
WIS eae = 
Hence we see that the cross-ratio of four points on a line is 
unaltered by a projective transformation of the points on the 
line. 
THEOREM 5. A projective transformation of the points on a line 
leaves invariant the cross-ratio of any four points on the line. 
10. Resultant of Two Transformations.—Let T and T, be 
two transformations whose equations are respectively 
_ ae+b * — am +h 
Ls ex¢-tad and #, = ci +d)" (1) 
The first transforms the point x into x, and the second 
transforms x, into z,.. We suppose the operations are carried 
out in the order in which the equations are written. If we 
eliminate x, from the above, we get 
70h (aa-+bic) « + (aib+bhid) 9 
“2 (e1a+dic) x + (cib+did) * (12) 
It should be observed that (12) is of the same form as (1) and 
differs from it only in the values of the coefficients. Equa- 
