447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 225 
only the surfaces X = 0, Y = 0, Z = 0, W =0 have a common intersection equivalent to 
n 3 — 1 points of intersection, but not equivalent to a complete common intersection of 
n 3 points. The last-mentioned circumstance would arise, if the condition of the common 
intersection should impose upon the surface more than ^ (n + 1) (n + 2) (rc + 3) — 4 con 
ditions ; viz., the surfaces would then be connected by an identical equation or syzygy 
olX + ¡3 Y + ryZ + 8 W =0. The common intersection is a figure composed of points and. 
curves: say it is the principal system in the first space; the problem is, to determine 
a principal system equivalent to n 3 — 1 points of intersection but such that the number 
of conditions to be satisfied by a surface passing through it is not more than 
£ {n + 1) (n + 2) (n + 3) — 4. 
75. The following locutions are convenient. We may say that the number of 
conditions imposed upon a surface of the order n which passes through the common 
intersection is the Postulation of this intersection; and that the number of points 
represented by the common intersection (in regard to the points of intersection of any 
three surfaces each of the order n which pass through it) is the Equivalence of this 
intersection. The conditions above referred to are thus 
Equivalence — n 3 — 1, 
Postulation ^ (n + 1) (n + 2) {n + 3) — 4. 
76. It would appear by the analogy of the rational transformation between two 
planes, that the only cases to be considered are those for which 
Postulation = £ (n + 1) (n + 2) (n + 3) — 4; 
but I cannot say whether this is so. 
77. In the transformation between two planes, the two conditions lead, as was 
seen, to the result that the curve aX + bY + cZ = 0 is unicursal. I do not see that 
in the present case of two spaces, the two conditions lead to the corresponding' result 
that the surface aX + bY + cZ+ dW = 0 is unicursal; that this is so, appears, however, 
at once from the general notion of the rational transformation. In fact, the equation 
in question aX + bY + cZ+dW = 0 is satisfied by x : y : z : w = X' : Y' : Z' : W and 
ax’ + by' -I- cz' + dw' = 0 ; the last equation determines the ratios x' : y' : / : w' in terms 
of two arbitrary parameters (say these are x : y' and x : z'), and we have then 
x : y : z : w proportional to rational functions of these two parameters; that is, the surface 
aX + bY + cZ + dW = 0 is unicursal. And similarly the surface aX' + bY' + cZ' + dW' = 0 
is unicursal. 
78. In the most general point of view, the principal system will contain a given 
number of points which are simple points, a given number which are quadriconical 
points, a given number which are cubiconical points, &c. &c., on the surfaces; and 
similarly a given number of curves which are simple curves, a given number which 
are double curves, &c. &c., on the surfaces. But, to simplify, I will consider that it 
includes only points which are simple points, and a curve which is a simple 
C. VII. 29 
curve