396 ON THE DEFICIENCY OF CERTAIN SURFACES. [524 
The formula therefore gives 
D = — ^ {n — 1) (n — 2) + 8 + k, 
viz. this is equal to the deficiency of the plane sections taken negatively. 
I find that the same property exists first in the case of a scroll (skew surface) 
having only a double curve; and secondly in the case of a torse (developable surface) 
having a cuspidal curve with the ordinary singularities; and this being so there can 
I think be no doubt but that it is true for any scroll or torse whatever—viz. that 
for any ruled surface whatever the deficiency is equal to that of the plane section 
taken negatively. 
First, for the scroll, we have 
D = i. (n — 1) (n — 2) (n — 3) — (n — 3) b + \q + 21, 
which should be 
= — £ (n — 1) (n — 2) + b. 
Salmon’s equations give in the case of a scroll 
St = (n — 4) {35 — n(n— 2)}, 
q = n (n — 2)(n — 5) — 2(n — 6) b, 
and with these values the relation is at once verified. 
Secondly, for the torse : changing the notation into that used for the singularities 
of the curve and torse, we have 
D = i (r - 1) (r - 2) (r - 3) - (r - 3) (x + m) + %(q + r) + 2t + f/3 + fy + a, 
which should be 
= — \ (m — 1) (m — 2) + h + /3. 
We have q = r(n — 3)—3a, and substituting this value and expressing everything in 
terms of r, m, n by means of the formulas 
os = h (r 2 — r — n — 3to), 
a = m — Sr + 3 n, 
i3 = n — Sr + 3m, 
t — ^ {r 3 — Sr 2 — 58r — Sr (n + 3m) + 42n + 78m}, 
7 = rm + 12r — 14m — Qn, 
h = ^ (m 2 — 10m — Sn + 8r), 
we have after all reductions 
D = — £ (m + n) + r — 1 = — f(m — 1) (m — 2) + h + ft.