342 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
[411 
41. We ought then to have 
Ъ (n - 2) = p, 
c (n — 2) = 2cr + в ; 
2 (q) + Si +j =2 p, 
3 (r) + c + 2i + ^ = 5a + 40 ; 
the first two of which give p, a, and then, substituting their values, the other two 
equations should become identities. In fact, attending to the values pr = 6, pq = c, the 
equations become 
26 (p + r — 2) + Sbq 4- 6 (/+ g) = 26 (n — 2), 
3c (p + q — 2) + c + 2cr + 6 (g + 3/) = f {c (n — 2) — с/} + 4c/. 
The first of these is 
2n = 2p + 2r + Sq+f + g, = (2ja +/) + (2r + Sq + g), 
and the second is 
fn = Sp + 3^ + 2r + g + I/, =| (2p +/) + (2r + 3 q + g), 
so that the equations are satisfied. 
Article No. 42. The Flecnodal Curve. 
42. A point on a surface may be flecnodal, viz. the tangent plane may meet the 
surface in a curve having at the point a flecnode, that is, a node with an inflexion 
on one of the branches. Salmon has shown that, for a surface of the order n without 
singularities, the locus of the flecnodal points, or flecnodal curve, is the complete 
intersection of the surface by a surface of the order lln —24, which may be called 
the flecnodal surface, the order of the curve being thus =n(lln — 24). I have 
succeeded in showing, in a somewhat peculiar way by consideration of a surface of 
revolution, that if the surface of the order n has a nodal curve of the order 6, and 
a cuspidal curve of the order c, then that the order of the flecnodal curve is 
= n(lln — 24) — 226 — 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) — 86 — 11c. 
Article Nos. 43 to 47. Surfaces of Revolution, in connexion with the Spinode Curve 
and the Flecnodal Curve. 
43. Consider a plane curve of the order m with 8 nodes and k cusps, and let 
this be 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