196 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
16. If ad — bc = 0, the equation of correspondence becomes 
(ax + by) (a'x' + b'y') = 0, 
and as before, to a single given point ax + by = 0, considered as belonging to the first 
figure, there corresponds every point whatever of the line, or second figure: to a 
single given point a'x' + b'y' = 0 (the same as, or different from, the first point), 
considered as belonging to the second figure, there corresponds every point whatever 
of the line, or first figure. 
17. Excluding the foregoing case, or assuming ad — be ^ 0, there are in general on 
the line two points such that to each of them considered as belonging to either 
figure there corresponds the same point considered as belonging to the other figure, 
or say there are two united points: in fact, writing x' : y' = x : y, we find 
ax 2 + (b + c)xy + dy 2 = 0, a quadric equation for the determination of the points in 
question. Unless 4ad — (b + c) 2 = 0, this equation will have two unequal roots; and 
taking the two points so determined for the fixed points A = A', B—B', the equation 
of correspondence will assume the form xy' — lex'y = 0. In this equation Jc cannot be = 1; 
for if it were so, the equation would be xy' — x'y = 0; that is, the points P, P' would 
be always one and the same point. The equation may, however, be xy' + x'y = 0; the 
points P, P' are then harmonics in regard to the fixed points A, B. It is to be 
observed, that if the equation xy' - kx'y = 0 be unaltered by the interchange of (x, y) 
and (x, y') we must have № — 1 = 0, or since = 1 is excluded, we must have k = — 1. 
18. The original equation (ax + by) x'+ (ex + dy) y'= 0. is unaltered by the inter 
change, only if b — c — 0; the equation 4ad — (b + cf = 0 becomes in this case ad — bc = 0, 
which by hypothesis is not satisfied; the two distinct points A = A', B = B' consequently 
exist. That is, if the correspondence between the two points P, P' is such that 
whether P be considered as belonging to the first figure or to the second figure, 
there corresponds to it in the other figure the same point P'—or say if the 
correspondence between the points P, P' is a symmetrical correspondence—then as 
united points in the superimposed figures we have the two distinct points A, B: 
and the correspondence of the points P, P' is given by the condition that these are 
harmonics in regard to the points A, B. 
19. There is still the case to be considered where 4ad — (b + cf — 0; the equation 
ax?+ (b + c) xy + dy* = 0 has here equal roots, or the two united points coincide 
together, or form a single point. Taking this point to be the point A, the coordinate 
whereof is x : y — 0 : 1, we must, it is clear, have d = 0, and therefore also b + c = 0 : 
the relation between the coordinates (x, y) and (x\ y) is then axx + b (xy' — x'y) = 0; 
viz., this is the form assumed by the equation of correspondence when instead of two 
united points there is a double united point, and this is taken to be the fixed 
point A. 
20. It is to be observed, that we cannot have either b = 0, for this would give 
xx — 0, which belongs to the excluded case ad — be = 0; nor a = 0, for this would give 
xy' — x'y — 0: excluding these cases, the equation is of necessity altered by the inter-