'234 
ON THE THEORY OF RECIPROCAL SURFACES. 
[750 
615. The question of singularities has been considered under a more general 
point of view by Zeuthen, in the memoir “ Recherche des singularités qui ont rapport 
à une droite multiple d’une surface,” Math. Annalen, t. IV. (1871), pp. 1—20. He 
attributes to the surface : 
A number of singular points, viz. points at any one of which the tangents form 
a cone of the order y, and class v, with y + y double lines, of which y are tangents 
to branches of the nodal curve through the point, and z + £ stationary lines, whereof 
z are tangents to branches of the cuspidal curve through the point, and with u double 
planes and v stationary planes ; moreover, these points have only the properties which 
are the most general in the case of a surface regarded as a locus of points; and 2 
denotes a sum extending to all such points. (The foregoing general definition includes 
the cnicnodes y = v — 2, y = y = z = Ç=ii = v = 0, and the binodes y — 2, y = 1, 
v = y = &c. = 0.) 
And, further, a number of singular planes, viz. planes any one of which touches 
along a curve of the class y and order v, with y' + y double tangents, of which y 
are generating lines of the node-couple torse, z + stationary tangents, of which z 
are generating lines of the spinode torse, v! double points and v cusps ; it is, more 
over, supposed that these planes have only the properties which are the most general 
in the case of a surface regarded as an envelope of its tangent planes; and 2' denotes 
a sum extending to all such " planes. (The definition includes the cnictropes y=v' — 2, 
y' = y = z = = u' = v' = 0, and the bitropes y = 2, y — 1, v = ÿ = &c. = 0.) 
616. This being so, and writing 
x = v + 2y + 3^, x’ — v + 2y + 3£", 
the equations (7), (8), (9), (10), (11), (12), contain, in respect of the new singularities 
additional terms, viz. these are 
a (n — 2) = ... + % \x (y — 2) — y — 2£], 
b (n - 2) = ... + S [y (y - 2)], 
c (n - 2) = ... + 2 [z (y, - 2)], 
a (n — 2) (n — 3) = ... + 2 \oc (— 4/a + 7) + 2y 4- 4£], 
b (n — 2) (n — 3) = ... + 2 \y (— 4/a + 8)] — 2' (4il + 3?/), 
c (n - 2) (n - 3) = ... + 2 [z (- 4/a + 9)] - 2' (2v'), 
and there are of course the reciprocal terms in the reciprocal equations (18), (19), 
(20), (21), (22), (23). These formulae are given without demonstration in the memoir 
just referred to: the principal object of the memoir, as shown by its title, is the 
consideration not of such singular points and planes, but of the multiple right lines 
of a surface; and in regard to these, the memoir should be consulted.