524] 
ON THE DEFICIENCY OF CERTAIN SURFACES. 
395 
I found in this manner the expression for the deficiency of a surface n having 
a double and cuspidal curve and the other singularities considered in my “ Memoir on 
the Theory of Reciprocal Surfaces,” Phil. Trans, vol. clix. (1869), [411, Coll. Math. 
Papers, vol. vi. p. 356]; viz. this was 
P = i(n-l)(n-2)(n- 3) - (n - 3) (b + c) + \{q + r) + 2t + 4- f y + i - $0, 
where we have 
h, order of double curve, 
q , class of Do., 
c , order of cuspidal curve, 
r, class of Do., 
¡3, number of intersections of the two curves, stationary points on b, 
y, number of intersections, stationary points on c, 
i , number of intersections, not stationary points on either curve, 
6, number of certain singular points on c, the nature of which I do not com 
pletely understand; it is here taken to be = 0. 
Before going further I remark that 
Postulation of right line qua ¿-tuple on surface n 
= (i + 1) n — (i + 1) (2i — 5), 
= ^i (i + 1) (3n — 2i + 5). 
Whence if a surface n has an ¿-tuple right line, the deficiency-value hereof is 
= (i — 1) (3n — 2¿ — 5), 
or we have 
D = ±(n- 1) (n — 2) (n — 3) — (i — 1)(3n — 2¿ — 5) 
= %(i — n + l)(i—n+2) (2i + n — 3) ; 
so that D = 0 if either i = n— 1 or i = n — 2; the former case is that of a scroll 
(skew surface) with a (n — 1) tuple right line, the latter that of a surface with a 
(ft — 2) tuple line: whence (as shown by Dr Noether) such surface is rationally trans 
formable into a plane. 
For a surface of the order n with an ¿-conical point where the tangent cone has 
8 double lines and k cuspidal lines, we have 
D = i(n — 1) (n — 2) (n — 3) — — 1) — 2) + (¿ — 2) (w — ¿ — 1) (S + «)} 
= ^ (n — i — 1) (n 2 + n — 5) + ¿ 2 — 4¿ -f 6 — 6 (¿ — 2) (8 + a:)} ; 
viz. for ¿ = n—l this is D = 0 (in fact, a surface n with a (n — 1) conical point is at 
once seen to be rationally transformable into a plane): and for ¿ = n, that is, for a 
cone of the order n, we have 
D = - \ (n — 1) (n — 2) + (77 - 2) (8 + k) - {n — 3) (8 + k), 
where the last term — (n — 3)(S + k) is added because in the present case the surface 
has the 8 double lines and the k cuspidal lines. 
50—2