192 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
section: I say equivalent to n 3 — 1 points, for this fixed intersection need not be 
n 3 — 1 points, but it may be or include a curve of intersection ( 1 ). The surfaces 
X = 0, F = 0, Z = 0, TF = 0 must consequently have a common intersection equivalent 
to n 3 — 1 points; there is (as in the preceding case) a cause of failure to be guarded 
against, viz., the condition as to the intersection must not be such as to imply 
one more point of intersection, that is, to imply an identical equation or syzygy 
aX + /3F + yZ + 8 TF = 0 between the functions X, F, Z, TF; but it is assumed that 
they are not thus connected. There is, then, a single variable point of intersection of 
the surfaces x' : y' : z : w = X : F : Z : TF; or taking the coordinates of this point 
to be (x, y, z, w), we have the ratios x : y : z : w rationally determined; that is, we 
have a second set of equations x : y : z : w = X' : Y' : Z' : TF', where X', F', Z', TF' 
are rational and integral functions of the same order, say n\ in the coordinates 
(x, y\ z, w'); viz., we have the rational transformation, as above, between the two 
spaces. 
6. Suppose that the common intersection of the surfaces X = 0, F = 0, Z = 0, TF = 0 
is or includes a curve of the order v; and consider in the first figure the two surfaces 
aX + /3Y + yZ + STF = 0, oqX + /3jF + y X Z + 8 X TF = 0, 
and the arbitrary plane ax + by + cz + dw = 0. The two surfaces intersect in the fixed 
curve v, and in a residual curve of the order n 2 — v; hence the two surfaces and the 
plane meet in v points on the fixed curve, and in n 2 — v other points. Corresponding 
to the surfaces and plane in the first figure, we have in the second figure the two 
planes 
ax' + (3y' + yz + 8w' = 0, a x x + ^yf + 7 1 z' + Sjw' = 0, 
and the surface aX’ + b Y' + cZ' + d W' = 0 of the order n': these intersect in n' points, 
being a system corresponding point to point with the n 2 — v points of the first figure; 
that is, we must have n' = n 2 — v. And conversely, it follows that in the second figure 
the common intersection of the surfaces X' =0, F = 0, Z' = 0, W — 0 will be or include 
a curve of the order v ; and that we shall have n = n 2 — v. Hence also 
v — v = (n — n') (n + n' + 1). 
7. The principle of the rational transformation comes out more clearly in the 
foregoing two cases than in the case of two lines, which from its very simplicity fails 
to exhibit the principle so well; and I have accordingly postponed the consideration of 
it: but the theory is similar to that of the foregoing cases. We must have the 
two sets (each a single equation) x : y = X : F, and x : y = X' : Y'. The equation 
x' : y' = X : Y must give for the ratio x : y a single variable value ; viz., there must 
be n — 1 constant values (values, that is, independent of x', y'); this can only be the 
case by reason of the functions having a common factor M of the order n — 1; but 
this being so, the common factor divides out, and the equation assumes the form 
x : y' = X : F, where X, F are linear functions of (x, y): and we have then reciprocally 
1 The curve of intersection may consist of distinct curves, each or any of which may be a singular 
curve of any kind in regard to the several surfaces.