214 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
that the order of the flecnodal curve is —n(lln— 24)— 22£ — 27c ; before giving this 
investigation, I will by the like principles demonstrate the above-mentioned theorem 
that the order of the spinode curve is =4n(n— 2)— 8Z>— 11c. 
Surfaces of Revolution, in connexion with the Spinode Curve and the Flecnodal Curve. 
Article Nos. 43 to 47. 
43. Consider a plane curve of the order m with & nodes and z cusps, and let this he 
made to revolve about an axis in its own plane, so as to generate a surface of revolution. 
The complete meridian section is made up of the given curve and of an equal curve 
situate symmetrically therewith on the other side of the axis ; the order of the surface 
is thus =2 m. 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 \{rrd—m) circles, nodal curves on the surface; the nodes generate 
<5 circles, which are nodal curves on the surface, and the cusps generate z circles, cus- 
pidal curves on the surface. There are m 2 — m — 2o — 3z circles of plane contact corre- 
sponding 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 ; yiz. these are the circular planes x-\-iy= 0, x—iy= 0 passing 
through the axis. I assume without stopping to show it that these 2 m lines are lines 
not/ but that is, that they each reduce the order of the spinode curve by 3*'. The 
inflexions generate 3m 2 — 6m— 65— 8z 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 TIessian=4(2m— 2), 
whence order of intersection = 16m 2 — 16m 
Nodal curve, \{nd‘ — m)+S circles, 8 times 8m 2 — 8m+16£ 
Cuspidal curve, z circles, 11 times ~\-22z 
Circles .of contact m 2 — m— 2£— 3z, 2m 2 — 2m— 4£— 6« 
Lines 2m , 3 times + 6m 
Spinode curve, 3m 2 — 6m— 6^— 8^ circles, 6m 2 — 12m— 12&— IQz 
16m 2 — 16m 
45. 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; 
* Observe that the terms in m cannot he got rid of in a different manner, by any alteration of the numbers 
8 & 11 to which the present investigation relates. 
