226 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
[447 
on the surfaces : this curve may, however, break up into separate curves, and we thus, 
in fact, admit the case where there are any number of separate curves each of them 
a simple curve on the surfaces. It is right to remark that we cannot assert a priori— 
and it is not in fact the case—that the principal system in the second space will be 
subject to the like restrictions: starting with such a principal system in the first 
space, we may be led in the second space to a principal system including a curve 
which is a double curve on the surfaces: an instance of this will in fact occui\ 
79. It is shown (Salmon’s Solid Geometry, 2nd ed., p. 283, [Ed. 4, p. 321]), that in 
the intersection of three surfaces of the orders /z, v, p respectively, a curve of inter 
section of the order m and class r counts as m (/z + v + p — 2) — r points of intersection. 
For a curve without actual double points or stationary points, we have r = m (m — 1) — 2h, 
where h is the number of apparent double points; or, substituting, we have the curve 
counting for to (/z + v + p — 2) — in (to — 1) 4- 2h points of intersection ; this is in fact a 
more general form of the formula, inasmuch as it extends to the case of a curve with 
actual double points and stationary points. Or, what is the same thing, the three 
surfaces intersecting in the curve of the order m with h apparent double points, will 
besides intersect in yvp - m(y + v +p — 2) + m (to — 1) — 2h points; viz., the curve may, 
besides the apparent double points, have actual double points and stationary points; 
but these do not affect the formula. 
80. Some caution is necessary in the application of the theorem. For instance, 
to consider cases that will present themselves in the sequel: let the surfaces be cubics 
(p, = y = p = 3); the number of remaining intersections is given as = 27 — 7m + m (to — 1)— 2h. 
Suppose that the curve consists of four non-intersecting lines, m = 4, h = 6, the number 
is given as = — 1. But observe in this case there are two lines each meeting the 
four given lines ; that is, any cubic surface passing through the four given lines meets 
these two lines each of them in four points, that is, the cubic passes also through 
each of the two lines; the complete cwn;e-intersection of the surfaces is made up of the 
six lines m = 6, h = 7 (since each of the two lines, as intersecting the four lines, gives 
actual double points, but the two lines together give one apparent double point), 
and the expression for the number of the remaining points of intersection becomes 
= 27 — 42 + 30 — 14 = 1, which is correct. 
81. Similarly, if the given curve of intersection be a conic and two non-intersecting 
lines, there is here in the plane of the conic a line meeting each of the two given 
lines, and therefore meeting the cubic surface, in four points, that is, lying wholly in 
the cubic surface: the complete crorve-intersection consists of the conic, the two given 
lines, and the last-mentioned line, m = 5, h = 5, and the number of points of intersection 
1S = 27 — 35 + 20 —10, = 2, which is correct. Again, if the given curve of intersection 
be two conics, here the line of intersection of the planes of the conics lies in the 
cubic surface; or, for the complete cizrye-intersection we have m = 5, h = 4; and the 
number of points is 27 — 35 + 20 — 8, =4. If in this last case each or either of the 
conics become a pair of intersecting lines, or if in the preceding case the conic becomes 
a pair of intersecting lines, the results remain unaltered.