PEOFESSOE CAYLEY ON EECIPEOCAL SUBFACES. 
221 
section of the three surfaces is 
=q,vg—m(flyqj + yuv + a(3g—2a(3y) + ccftyr. 
Apply this to the case of a surface of the order n with a nodal curve of the order 5 and 
class ?, intersecting the Hessian and flecnodal surfaces, we have 
Order. Passing through ( b , ?), times 
Surface n 2 
Hessian 4 n — 8 4 
Flecnodal lln— 24 11 
whence number of intersections is 
= 4n(n-2){lln - 24) - b {n . 4 . 1 1 + (4 n - 8)11 . 2 + (lln - 24)2 . 4 - 2 . 2 . 4 . 11 } 
+2.4.11?, 
that is 
=4n(n- 2)(lln- 24) - (220 n- 544)5- 88?; 
and the value of ft' is one half of this, 
==2?2(w—2)(llw— 24)— (llOw— 272)5+44?. 
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 — 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 g ; the surfaces v intersect in the curve m a. times, and in a residual curve of the 
order [iv—mu, this last meets the surface f in %([Lv—ma) points, and thence the three 
surfaces meet in pyg—mug — I points. But since m is a simple curve on each of the 
surfaces v, g>, the three surfaces meet in — m) — a\qn(v-\- p—2) — r] points, whence 
equating the two values 
— 2a) — ax. 
Next, let the curve m be a-tuple on the surface [Jj, /3-tuple on the surface Considering 
the new surface § through m, then p, v intersect in the curve m a/3 times, and in a resi- 
dual curve of the order (mv — mufi; this last meets the surface g in gfav — ma/3) points; 
whence the three surfaces meet in g(p— ??za/3) — I points. But the curve m being a 
/3-tuple curve on v, and a simple curve on these meet in the curve m /3 times and in a 
residual curve of the order vg — /3m, whence the three surfaces meet in 
qj(yg— pm)— a[m(v+/3 ? — 2/3)— /3r] 
points ; and equating the two values, we have 
I=m((3qj + ay — 2 a/3) — 2a/3r. 
Lastly, if the curve m be y-tuple on g, then the surfaces q,, g meet in m ay times and 
in a residual curve of the order qg—may ; this last meets v in 
y(qg — aym) — (3 \m(ya +ac — 2 ay) — ayr] 
