416] ON THE THEORY OF RECIPROCAL SURFACES. 591 
635. 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. pp. 1—20, 1871. 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 + rj double lines, of which y are tangents 
to branches of the nodal curve through the point, and z + f stationary lines, whereof 
^ 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 = 7) = z = Ç = u = v = 0), and [also, but not properly] the binodes (y = 2, 
y = l, v = y — &c. = 0), [it includes also the off-points (y = v = 3, z — v = 1, y = ’7 = (=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 + rf 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, u' double points and v' cusps ; it is, moreover, 
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 = rj' = z' = £' = v! = v' = 0), and [also, but not properly] the bitropes (y =2, = 1, 
v = y' = &c. = 0), [it includes also the off-planes (y = v = 3, z —v' — 1, y = y = Ç' = 0)].} 
636. This being so, and writing 
x = v + 2tj + 3£, x' = v + 21] + 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 \cc (fi — 2) — y — 2f], 
b (n —2) = ... + %[y(fi — 2)], 
c (n — 2) = ... + 2 [z (y, — 2)], 
a(n — 2) (ii — 3) = ... + 2 \_x (— 4y, + 7 ) + 2t7 + 4£], 
h (n — 2) (n — 3) = ... + 2 \y (— 4/x + 8)] — 2' (4u' + Sv'), 
c (n - 2)(n - 3)= ... + 2 [z (- 4/x + 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 consider 
ation 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.