[447 
447] 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
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82. If a surface of the order yu, pass through a curve of the order m and class 
r without stationary points or actual double points, this imposes on the surface a 
number of conditions = (yu, +1) m — \r. In the case in question, the value of r is 
= m (to — 1) — 2h ; or, substituting, the number of conditions is =(ya+ l)m — \m(m— l)+/i ; 
and the formula in this form holds good even in the case where the curve has 
stationary points and actual double points. Thus /a = 3, the number of conditions is 
= — — + If the curve be a line, m= 1, h = 0, number of conditions is 
= 4; if the curve be a pair of non-intersecting lines, m = 2, h = 1, number of con 
ditions is = 8. And so in general, if the curve consist of k non-intersecting lines 
(k = 4 at most), then m = k, li = \k(k— 1), and the number of conditions is = 47c. If 
the curve be a conic, or a pair of intersecting lines, m = 2, h — 1, and the number of 
conditions is = 7. If the curve consist of k lines, such that there are 6 pairs of inter 
secting lines, then m = k, h = ^k (Jc — 1) — 6, and the number of conditions is =4k—6. 
It is obvious that, the number of conditions for a line being = 4, that for the k lines 
with 6 intersecting pairs must have the foregoing value 4k — 6. In fact, when the 
lines do not intersect, we take on each line 4 points, and the cubic surface passing 
through any such 4 points will contain the line; but for two lines which intersect, 
taking this point, and on each of the intersecting lines 3 other points, the cubic 
surface through the 7 points will pass through the two lines; and so in other cases. 
83. The formula must, in some instances, be applied with caution. Thus, given 
five non-intersecting lines k = 5, 0 = 0,* and the number of conditions is =20; and a 
cubic surface cannot be, in general, made to pass through the lines. But if the five 
lines are met by any other line, then a cubic surface, if it pass through the five 
lines, will pass through this sixth line; for the six lines k = 6, 6 — 5, and the number 
of conditions is 24 — 5 = 19 ; so that there is a determinate cubic surface through the 
six lines, and therefore through the five lines related in the manner just referred to. 
84. Recurring to the problem of transformation, it appears by what precedes, that 
if the principal system in the first plane consists of a x points, and of a curve of the 
order m x with h x apparent double points (the ol x points being simple points, and the 
curve a simple curve on the surfaces), then the conditions for a transformation are 
(3n — 2) m x — m x (m x — 1) + 2h x + a x = n 3 — 1, 
( n + 1) m x — ^m x (m x — 1) + h x + a x = % (n + 1) (n + 2) (n + 3) — 4, 
where, in the second line, instead of I have written =. I remark, in passing, 
that I have ascertained that an actual triple point counts as an apparent double 
point; or, what is the same thing, that if the curve has t x actual triple points, then 
we may, instead of h X) write h x + t x . The equations give 
m x (4n — 5 — ^ (n — 1) (5?i 2 — n — 12) — 2h x , 
(n — 4) m x — a = -¡I (n — 1) (2n 2 — 4n — 15), 
to which may be joined 
(3n + 8) m x - 2m x (rn x — 1) + 4h x + oa x = (n - 1) (6n + 17). 
29—2