416] 
ON THE THEORY OF RECIPROCAL SURFACES. 
585 
approach together, and, when the plane passes through the cusp, unite into a singular 
point in the nature of a triple point (= node + two cusps) ; and when the plane passes 
below the cusp, the two cusps of the section become imaginary, and the nodal line 
changes from crunodal to acnodal. 
625. At a point i the nodal curve crosses the cuspidal curve, being on the side 
away from the two half-sheets of the surface acnodal, and on the side of the two 
half-sheets crunodal, viz. the two half-sheets intersect each other along this portion of 
the nodal curve. There is at the point a single tangent plane, which is a plane i' ; 
and we thus have i = i. 
626. As already mentioned, a cnicnode G is a point where, instead of a tangent 
plane, we have a tangent quadricone ; and at a binode B the quadricone degenerates 
into a pair of planes. A cnictrope C is a plane touching the surface along a conic ; 
in the case of a bitrope B', the conic degenerates into a flat conic or pair of points. 
627. In the original formulae for a (n — 2), 6 (ft — 2), c (n — 2), we have to write « — B 
instead of k, and the formulae are further modified by reason of the singularities 6 
and to. So in the original formulae for a (n — 2) (n — 3), b (n — 2) (n — 3), c (n — 2) (n — 3), 
we have instead of 8 to write 8 — G — Sto ; and to substitute new expressions for 
[ab], [ac], [be], viz. these are 
[ab] = ab-2p - j, 
[ac] = ac — 3cr — % — to, 
[¿>c] = be — 3/3 — 27 — i. 
The whole series of equations thus is 
(1) 
(2) 
(S) 
(4) 
(5) 
(6) 
0) 
(8) 
(9) 
(10) 
(11) 
(12) 
(13) 
(14) 
a = a. 
/'=/• 
%' = i. 
a = 11 (11 — 1) — 2 b — 3c. 
k! = 311 {n — 2) — 6 b — 8c. 
8' = \n (n — 2) (?i 2 — 9) — (w 2 — n — 6) (26 + 3c) + 26 (6 — 1 ) + 66c + fc (c — 1). 
a (n — 2) = k — B + p + 2cr -|- 3(w. 
6 (n — 2) = p -f- 2/3 + 37 + St. 
c(n— 2)= 2a + 4/3 + 7 + 0 + &). 
a (n — 2) (11 — 3) = 2 (8 — G — Sto) + 3 (ac — Sa — % — Sto) + 2 (ab — 2 p — j 
7./„ o\/„ o\ ai. _|_ ( a b —2p—j ) + S(bc— S/3 — 2y — i). 
+ (ac — 3o’ — % — 3to) + 2 (be — 3/3 — 2<y — i). 
)• 
6 (11 — 2) (n — 3) = 4/b 
c (w — 2) (ft — 3) = 6/1 
q ■= b~ — 6 — 2k — 2/— 37 
r — c~ - c — 2h — 3/3. 
-6*. 
C. VI. 
7 4