447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 191 
then starting with the set x : y' : z — X : Y : Z, for any given point (x, y, z) what 
ever in the first figure, we have a single corresponding point (xy', z) in the second 
figure; but for any given point (x, y, z) in the second figure, we have primd facie 
a system of n 2 points in the second figure, viz., these are the common points of 
intersection of the curves x' : y : / = X : Y : Z (in which equations x, y', z are 
regarded as given parameters, x, y, z as current coordinates, and the equations there 
fore represent curves of the order n in the first figure). The curves may however 
have only a single variable point of intersection; viz., this will be the case if each 
of the curves passes through the same n 2 — 1 fixed points (points, that is, the positions 
of which are independent of x', y', z'); and in order that the curves in question may 
each pass through the n 2 — 1 points, it is necessary and sufficient that these shall be 
common points of intersection of the curves X = 0, F = 0, Z=0. {Observe that the 
condition thus imposed upon the curves X = 0, F=0, Z = 0 will in certain cases 
imply that the curves have n 2 common intersections; or, what is the same thing, that 
the functions X, F, Z are connected by an identical equation, or syzygy, aX + ¡3Y + yZ = 0. 
This must not happen; for if it did, not only there will be no variable point of 
intersection, and the transformation will on this account fail; but there would also 
arise a relation ax' + ¡By' + yz' =0 between (x', y', z), contrary to the hypothesis that 
{x', y', z) are the coordinates of any point whatever of the second figure. It thus 
becomes necessary to show that there exist curves X = 0, F = 0, Z = 0, satisfying the 
required condition of the n 2 — 1 common intersections, but without a remaining common 
intersection, or, what is the same thing, without any syzygy aX + /3F+ yZ = 0.} 
3. The curves x : y' : / = X : F : Z having then a single variable point of 
intersection, if we take (x, y, z) to be the coordinates of this point, the ratios x : y : z 
will be determined rationally; that is, as a consequence of the first set of equations, 
we obtain a second set x : y : z — X' : Y' : Z', where X', Y\ Z' will be rational 
and integral functions of the same order, say n', of the coordinates (x', y', z); that 
is, we have a second set of equations, and consequently a rational transformation, as 
mentioned above. 
4. It is easy to see that we have n' = n; in fact, consider in the first figure 
a curve aX + /3 F + yZ = 0, and an arbitrary line ax + by + cz = 0 ; to these respec 
tively correspond, in the second figure, the line ax' + ¡By' + yz — 0, and the curve 
aX' + bY' 4- cZ 1 = 0; the curves are of the orders n, n' respectively, or the curve and 
line of the first figure intersect in n points, and the line and curve of the second 
figure intersect in n points; which two systems of points must correspond point to 
point to each other; that is, we must have n' = n. It will presently appear how 
different the analogous relation is in the transformation between two spaces. 
5. Ascending to the case of two spaces, we have here the two sets 
x' : y' : z' : w'= X : Y : Z : W; x : y : z : w = X' : Y' : Z' : W', 
the theory is analogous; the surfaces x : y' : z : w' = X : F : Z : IF (surfaces of the 
order n in the first figure) must have a single variable point of intersection, and they 
must therefore have a common fixed intersection equivalent to n 3 4 5 — 1 points of inter-