198 ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
X = 0, F = 0, Z = 0 must have a common intersection of n- — 1 points, and no more; 
that is, they must not have a complete common intersection of n 2 points. In the case 
n — 2, taking the n 2 — 1 points in the first plane to be any three points whatever, the 
condition that the curves shall be conics passing through the three points does not in 
anywise imply that the conics shall have a common fourth point of intersection; and 
we have thus a rational transformation as required; viz., the first set of equations is 
x' : y' : z = X : Y : Z, where X = 0, Y = 0, Z = 0 are conics passing through the same 
three points of the first plane ; and as it is easy to see (but which will be subsequently 
shown more in detail), the second set is the similar one x : y : z = X' : Y' : Z', where 
X' = 0, Y' = 0, Z' = 0 are conics passing through the same three points in the second 
plane; this may be called the quadric transformation between the two planes. 
24. But the like theory would not apply to the case n = 3; if the n 2 — 1 points 
in the first plane were any eight points whatever, the cubics X = 0, F = 0, Z — 0, 
intersecting in these eight points, would have a common ninth point of intersection, 
and the transformation would fail; and so for any higher value of n, taking at pleasure 
any | n (n + 3) — 1 of the n 2 — 1 points of the first plane, the curves X = 0, F = 0, Z = 0 
of the order n passing through these ^n(n + 3) —1 points, would have in common all 
their remaining points of intersection, and the transformation would fail. A trans 
formation can only be obtained by taking the n 2 — 1 points in such wise that these 
can be made to be the common intersection of the curves, and at the same time that 
the number of conditions imposed upon each of the curves X = 0, F = 0, Z = 0 shall be 
at most = \n (n + 3)— 1. 
25. And this requirement may be satisfied; viz., the number of conditions may 
be made to be =^n(n + 3) — 1, by assuming that certain of the n 2 — 1 points of inter 
sections are multiple intersections of the curves. For if we have a given point which 
is an a-tuple point on each of the curves X = 0, F = 0, Z = 0, then this counts for 
a 2 points of intersection of any two of the curves, and thus for a 2 points of the n 2 — 1 
points: but the condition that the given point shall be on any one of the curves, 
say the curve X = 0, an a-tuple point, imposes on the curve, not a 2 , but only ia(a+l) 
conditions: and we have in this way a reduction whereby the number of conditions 
for passing through the n 2 — 1 points can be lowered from n 2 — 1 to the required number 
\n (n + 3) — 1. 
26. In particular, for n = 3, we may for the ri 2 — I points of the first plane take a 
point as a double point on each of the cubic curves X = 0, F = 0, Z= 0 (which therefore 
reckons as four points), and take any other four points. Each of the curves is determined 
by the conditions of having a given point for double point, and of passing through 
the same four other given points; that is, by 3+4=7 conditions ; and the three cubic 
curves X = 0, F= 0, Z— 0 have for the common intersection the double point reckoning 
as four points, and the given other four points; that is, they have a common inter 
section of 4 + 4 = 8 points; but this does not imply that they have a common ninth 
point of intersection ; we have therefore a rational transformation as required; viz., the 
first set of equations is x' : y' : / = X : Y : Z ; where X = 0, F = 0, Z = 0 are cubics 
in the first plane having each of them a double point at the same given point and