447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 197 
change of (x, y) and (x', y); that is, in the case of a double united point, the 
transformation is essentially unsymmetrical. 
By what precedes, if the other fixed point be taken to be at infinity, the coordi 
nates x : y and x : y' may be taken to be p, p respectively; viz., p, p will denote 
the distances of the points P, P' from the double united point A ; and the equation 
of correspondence then becomes pp + b (p — p) = 0 ; that is, (p — b) (// + b) + b 2 = 0. 
21. The original equation axx' + byx' + cxy + dyy' — 0 can be reduced to the 
inverse form xx' — yy' = 0 only (it is clear) in the symmetrical case b = c; here, trans 
forming to the united points, the equation is, by what precedes {ante, No. 17) xy+x'y = 0. 
This equation can be written {lx + my) {lx' 4- my) — {lx — my) {lx' — my') = 0, where l : m 
is arbitrary ; viz., we have thus an equation of the required form. 
22. In further explanation, start from the equation app + b {p + p) + d — 0 ; that 
is, {ap + b) {ap' +b) + ad — b 2 = 0, or say {p — a) {p - a) — k 2 — 0 ; this may be reduced to 
pp —1 = 0; viz., the point 0 from which are measured the distances p, p is here the 
mid-point between the two united points A, B; and the unit of distance is \AB\ 
the equation expresses that the points P, P', harmonics in regard to the two points 
A, B, are the images one of the other in regard to the circle described upon AB 
as diameter. Take any two corresponding points L, L'; if the distances of these be 
A, A', we have AA' = 1 ; and hence 
(p — A ) {p — A ) = 1 — A (p + p ) + A 2 = A (A+A p — p), 
(p - A') (p' - A') = l - A' (p + p') + A /2 = A' (A + A' — p — p') ; 
p — A p — A A 
p — A' p' — A' ’ 
x Jc (p — A) x' k {p — A) 
y Py P ~ x 
k 2 — ^-,{so that k 2 ^1); or, k = \ = —,, 
becomes xx' -yy' = 0; that is, the correspondence of the points P, P' being symmetrical, 
if the coordinate - of P be taken to be a multiple of the ratio of the distances 
y 
PL, PL' of P from any two corresponding points L, L' (and of course the coordinate 
—, of P' to be the same multiple of the ratio of the distances P'L, P'L'), the equation 
y 
of correspondence is obtained in the inverse form xx' — yy = 0. 
and consequently 
which, writing 
The Rational Transformation between Two Planes. 
23. Starting from the equations x : y' : z = X : Y : Z, where X — 0, Y= 0, Z=0 
are curves in the first plane, of the same order n, it has been seen that in order 
that we may thence have a rational transformation between the two planes, the curves