100 
PROFESSOR CAYLEY ON THE CURVES 
55. Now in every case except (5) m the number of intersections of the conic with the 
curve is >6 (viz. for (4, l) m and (3, 2) m the number is 7, for (3, 1, 1) and (2, 2, 1) it is 8, 
and for the remaining two cases it is 9 and 10 respectively) ; hence if the given curve 
m be a cubic, the number of conics satisfying the prescribed conditions is = 0 ; and since 
a cubic may be the general cubic or a nodal or a cuspidal cubic, we have the three cases 
(m, n, a)=( 3, 6, 18), (3, 4, 12), and (3, 3, 10). We have thus in each case three con- 
ditions for the determination of the constants a, c; so that there is in each case a veri- 
fication of the resulting formula. 
56. In the omitted case (5) m , when the curve mis a cubic, the theory of the conics 
(5) m is a known one, viz. the points of contact of these conics, or the “ sextactic” points 
of the cubic, are the points of contact of the tangents from the points of inflexion ; the 
number of the conics (5) m is thus =(n— 3)i, viz. in the three cases respectively it is =27, 
3, and 0. Hence for determining the constants we have the three equations 
9<z+18c=27, 
7a-\-12c= 3, 
6<z + 10c= 0, 
which are satisfied by a= — 15, c=9, and the resulting formula is 
(5)=— 15m— 15w+9a. 
In the particular case of a curve without nodes or cusps, this is (5)=12w— 15m, 
=m(12m— 27), which agrees with the result obtained in my memoir “On the Sextactic 
Points of a Plane Curve,” Phil. Trans, vol. civ. (1865) pp. 545-578. 
57. The subsidiary results required for the remaining cases (4, 1), &c. are at once 
obtained from the foregoing formulse for (4Z)(1), (3Z)(2), &c. ; for example, we have 
(4) m (l) m ,= n'( — 10m — 8 n-\-6a) 
+m f (— 8m— 10w+6a), 
with like expressions for (3, l) m (l) te -, &c., 
(3) m (2) TO , = (_8m-8rc+6 a ), 
(3) m (l, l) m , =(>' 2 -\d )(— 4m— 3^ + 3«) 
+ (m'«'- fa' )( — 8m— 8^+6 a) 
+(^m' 2 — |m')( — 3m— 4w+3a) ; 
with like expressions for (2, l) m (2) m ,, (2, l) m (l, l) m ,, &c. &c. 
58. Calculation of (4, 1). We have 
(4, 1 W-(4, l).-(4, 1)„,=(4)„(1),,+(4)„.(1)„ 
= — 16 mm' — 2 0 (md -f m'n) —IQnd 
+ 6 (a n'-\- a'n) ■+ 6(am'-{- a'm), 
the integral of which is 
( 4, l) m = — 8m 2 — 2 0 mn — Sn 2 + «(m + n ) + «( 6 m + 6n + c ) . 
