411] 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
343 
order of the surface is thus = 2to. The two curves intersect in m points on the axis 
and in m 2 — m points, forming \ (m 2 — m) pairs of points, situate symmetrically on 
opposite sides of the axes; these last generate ^ (m 2 — m) circles, nodal curves on the 
surface; the nodes generate 8 circles, which are nodal curves on the surface, and the 
cusps generate « circles, cuspidal curves on the surface. There are m 2 — m —28 -3« 
circles of plane contact corresponding in the plane curve to the tangents perpendicular 
to the axis. Each of the m points on the axis gives in the surface a pair of 
(imaginary) lines; and we have thus two sets each of m lines, such that along the 
lines of each set the surface is touched by an (imaginary) meridian plane; viz. these 
are the circular planes x + iy = 0, x — iy = 0 passing through the axis. I assume 
without stopping to show it that these 2m lines are lines not j' but that is, that 
they each reduce the order of the spinode curve by 3( x ). The inflexions generate 
3m 2 — 6m — 68 — 8« circles which constitute the spinode curve on the surface. 
44. And we can thus verify that the complete intersection of the surface with 
the Hessian is made up in accordance with the foregoing theory; viz. 
Order of surface = 2m, 
Order of Hessian = 4 (2m — 2), 
whence order of intersection = 16m 2 —16m 
Nodal curve, ^-(m 2 — m) + 8 circles, 8 times 8m 2 — 8?n + 168 
Cuspidal curve, « circles, 11 times + 22« 
Circles of contact m 2 — m — 28 — 3«, 2m 2 — 2in — 48 — 6« 
Lines 2m , 3 times + 6m 
Spinode curve, 3m 2 — 6m — 68 — 8« circles, 6m 2 — 12m — 128 — 16« 
16m 2 — 16m. 
4o. We may by a similar reasoning show that the surface and the flecnode surface 
intersect in the nodal curve taken 22 times, and in the cuspidal curve taken 27 times; 
and consequently that the order of the residual intersection or flecnodal curve is 
= n (lira - 24) — 22b — 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 fleflecnode. 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 fleflecnode circle; and I assume that it counts twice as a flecnode 
circle. Hence if the number of the points P be = 0, we have on the surface 0 
fleflecnode circles, = 20 flecnode circles, that is, a flecnode curve of the order 40. 
I wish to show that we have 0 = 5m 2 — 9m — 108 — 12«. 
1 Observe that the terms in m cannot be got rid of in a different manner, by any alteration of the 
numbers 8 and 11 to which the present investigation relates.