394 
[524 
524. 
ON THE DEFICIENCY OF CERTAIN SURFACES. 
[From the Mathematische Annalen, vol. III. (1871), pp. 526—529.] 
If a given point or curve is to be an ordinary or singular point or curve on a 
surface of the order n, this imposes on the surface a certain number of conditions, 
which number may be termed the “ Postulation ”; thus “ Postulation of a given curve 
qua ¿-tuple curve on a surface n” will denote the number of conditions to be satisfied 
by the surface in order that the given curve may be an ¿-tuple curve on the surface. 
The “ deficiency ” (Flachengeschlecht) of a given surface of the order n is 
= ±(n — l)(n — 2)(n — 3) less deficiency-value of the several singularities; viz. as shown 
by Dr Noether, if the surface has a given ¿-tuple curve, the deficiency-value hereof is 
= Postulation of the curve qua (¿ — 1) tuple curve on a surface n — 4; 
and if the surface has an ¿-conical point, the deficiency-value hereof is 
= Postulation of the point qua (i — 2) conical point on a surface n — 4; viz. this 
is = (i — 1) (i — 2), and is thus independent of the order of the surface. 
I remark that if the tangent-cone at the ¿-conical point has 8 double lines and 
k cuspidal lines, then the deficiency-value is 
= (i — 1) (i — 2) + (i — 2) (n — i — 1) (8 + k). 
In the case of a double or cuspidal curve ¿ is = 2, and the deficiency-value is 
= Postulation of given curve qua simple curve on a surface n — 4; 
and so for an ordinary conical point ¿ is = 2, and the deficiency-value is = 0: results 
which were first obtained by Dr Clebsch.