PEOEESSOE CAYLEY ON EECIPROCAL .SURFACES. 
215 
and consequently that the order of the residual intersection or fieenodal curve is 
=»(llw-24)— 225— 27c. 
To effect this, observe that at any point whatever of a quadric surface the tangent plane 
meets the surface in a pair of lines, that is, in a curve having at the point of contact a 
node with an inflexion on each branch, or say, a flefleenode. Imagine in the plane figure 
a conic having its centre on the axis of rotation and its axis coincident therewith, and 
the conic having with the curve of the order m a 4-pointic intersection at any point P ; 
the point P generates a circle, such that along this circle the surface is osculated by a 
quadric surface of revolution in such wise that the meridian sections have a four-pointic 
contact ; the circle in question is thus on the surface a flefleenode circle ; and I assume 
that it counts twice as a flecnode circle. Hence if the number of the points P be =Q, 
we have on the surface 0 flefleenode circles, = 20 flecnode circles, that is, a flecnode curve 
of the order 40. I wish to show that we have 4=5m 2 — 9m— 10&— 12z. 
46. The problem is as follows : given a curve of the order m with § nodes and z cusps ; 
it is required to find the number of the conics, centre on a given line, and an axis coin- 
cident in direction with this line, which have with the given curve a 4-pointic intersec- 
tion, or contact of the third order. This may be solved by means of formulae contained 
in my “Memoir on the Curves which satisfy given Conditions”*. 
Taking #=Q for the given line, the conic ( a , b , c,f, g , lijx, y , 1) 2 =0 will have its 
centre on the given line and an axis coincident therewith, if only 7i=0, g=0 ; and denoting 
these two conditions by 2X, it is easy to see that we have 
(2X.-.)=1, (2X:/)=2, (2X-//)=2, (2X///)=1. 
But in general if the conic satisfy any other three conditions 3Z, then the number of the 
conics (2X, 3Z) is 
= K \ — 47+-P) 
+/3 / (~t a +i^3+Ay~^ -p) 
+r'( i» ), 
where a, (3, y, & denote (2X.\), (2X:/), (2X • //), (2X///), viz. in the present case the 
values are 1, 2, 2, 1 respectively, and where a', (3','y' denote (3Z :), (3Z • /), (3Z//) 
respectively. 
47. Substituting for a, (3 , y, c5 their values, the number of the conics in question is 
=jz(3', that is =^(SZ • /). Suppose that 3Z, or say 3, denotes the condition of a contact 
of the third order with a given curve ( m , £>, z), or say with a given curve (m, n, a) ( m the 
order, n the class =m 2 — m— 2&— 3z, u=3n-\-z), then we have 
(3: )= — 4m— 3%+3«, 
(3 ;'/)= — 8 m — 8re-f-6a, 
(3// )= — 3m — 4?i+3a; 
and from the second of these the number of the conics in question is = — 4m — 4«-j-3ct, 
that is, it is = — 4m+5?i-|-3;f, or finally it is =5m 2 — 9m— 10&— 12z. 
* Philosophical Transactions, t. 158 (1868), pp. 75-144: see p. 88. 
