194 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
11. A special case is when 
Writing here 
ad — bc = 0. 
c d 
V 
- = - = m, = -, 
ab a 
the equation is 
that is 
(ax + by) ( x + my) = 0, 
(ax + by) (a'x + b'y) = 0 ; 
viz., either ax + by= 0, without any relation between x', y'; or else a'x! + b'y' = 0, 
without any relation between x, y; that is, to the single point ax + by = 0 of the first 
figure there corresponds any point whatever of the second figure; and to the single 
point a'x + b'y' = 0 of the second figure there corresponds any point whatever of the 
first figure. 
12. In the general case where ad — be ^ 0, we may either by a linear transformation 
(ax + by, cx + dy into y, — x or into x, — y) of the coordinates of a point of the first 
figure, or by a linear transformation (ax' + cy, bx +dy' into y, —x or into x, — y) of 
the coordinates of a point in the second figure (or in a variety of ways by simultaneous 
linear transformations of the two sets of coordinates) transform the relation indifferently 
into either of the forms xy — xy = 0, xx — yy' = 0 ; the former of these, or x' : y — x : y, is 
the most simple expression of the homographic transformation ; the latter, or x' : y' = - : -, 
X y 
is its expression as an inverse transformation. 
13. If, to fix the precise signification of the coordinates (x, y), we employ the 
distances from a fixed point 0 in the line; taking the distances of the two fixed 
points (say A, B) to be a, /3, and that of the variable point P to be p, then we 
have x, y proportional to given multiples jp(p-a), q(p—/3) of the distances from the 
two fixed points; or writing - = n, we may say that the coordinate - of the point 
^ y 
P is =n 
P-/3’ 
p — a. 
n we write 
If for 
or in particular, if n= 1, then the coordinate is =-— 
P — P 
an d then take /3 = oo, we see that in a particular system 
p-p p-P 
of coordinates, A at 0, B at oo, the coordinate - is = p. Proceeding in the same 
manner in regard to the coordinates (x', y), for a particular system of coordinates, 
A' at O', B' at oo, the coordinate —, of P' will be = p. And the correspondence 
of the points P, P' will be given by an equation 
app + bp q- cp d — 0. 
14. The equation just mentioned is often convenient for obtaining a precise statement 
of theorems. Thus taking A, B at pleasure on the first line, A', B' the corresponding 
points on the second line, we have 
cp + d