350 
[411 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
and the value of /3' is one half of this, 
= 2n (n - 2) (lira - 24) - (HOra - 272) b + 44g. 
I have not succeeded in applying the like considerations to the cuspidal curve. 
58. As regards the general theorem, we know (Salmon, p. 274) that if two surfaces 
of the orders p, v partially intersect in a curve of the order m and class r, and 
besides in a curve of the order m', then the curves m, m! meet in m (/a + v — 2) — r 
points. 
Suppose that the curve m is a-tuple on the surface p\ then to find the number 
I of the intersections of the curves m and m', we may imagine through m a surface 
of the order p; the surfaces p, v intersect in the curve m a times, and in a 
residual curve of the order pv — ma, this last meets the surface p in p (pv — ma) 
points, and thence the three surfaces meet in pvp — map — I points. But since 
m is a simple curve on each of the surfaces v, p, the three surfaces meet in 
p (vp —m) — a \in (v + p — 2) — r] points, whence equating the two values 
I = m(p + av — 2a) — an 
Next, let the curve m be a-tuple on the surface p, /3-tuple on the surface v. Con 
sidering the new surface p through m, then p, v intersect in the curve m aß times, 
and in a residual curve of the order pv — maß; this last meets the- surface p in 
p (pv — maß) points; whence the three surfaces meet in p (pv — maß) — I points. But 
the curve m being a /3-tuple curve on v, and a simple curve on p, these meet in 
the curve m ß times and in a residual curve of the order vp — /3m, whence the three 
surfaces meet in 
p (vp — ßm) — a [m (v + ßp — 2/3) — ßr] 
points; and equating the two values, we have 
I — m (ßp + av— 2aß) — 2aßr. 
Lastly, if the curve m be 7-tuple on p, then the surfaces p, p meet in m ay 
times and in a residual curve of the order pp — may; this last meets v in 
v (pp — aym) — ß [m (7p + ap— 2ay) — ay?•] 
points, that is, the number of points of intersection of the three surfaces is 
= pvp — m (ßyp 4- yav + aßp — 2aßy) + aßyr. 
59. I represent the complete value of ß' by 
ß' = 2ra(llra-24) 
- (110n - 272) b + 44q 
- (116/1- 303) c + ^r 
+ £f±/3 + 2487 + 198i 
- liG — gB — oci — \j — px — v6 
- It O' - g'B' - x’i! - \'j' - p'x' - v 0',