596 
NOTES AND REFERENCES. 
and assuming the correctness of Zenthen’s values it would seem to follow that the 
four forms of surface have 
12, 6, 12, 1 
actual double planes respectively. 
113. In the equation No. 36, il = AP + BQ + CR + .. = 0, it is implicitly assumed 
that the number of terms P, Q, R,.. is finite, viz. the implied theorem is that any 
given /¿-fold relation whatever (k of course a finite number) there is always a finite 
number of functions P, Q, R,... such that every onefold relation included in the P-fold 
relation is of the form in question il, = AP + BQ + CR +..., =0: this seems self- 
evident enough, but I never succeeded in finding a proof: a proof of the theorem has 
however been obtained by Hilbert, see his papers “ Zur Theorie der algebraischen 
Gebilden (Erste Note),” Gott. Nachr. No. 16, (1888), pp. 450—457. 
411, 415, 416. The first and second of these papers precede in date Zeuthen’s 
Memoir of 1871 referred to in 416, but I ought in that paper to have referred also 
to his later Memoir, “ Revision et extension des formules numériques de la théorie 
des surfaces réciproques,” Math. Ann. t. x. (1876), pp. 446—546. I compare the 
notations as follows, viz. for the unaccented letters we have 
Cayley. 
n, a, 8, k, p, a 
h, q, k, t, 7 
c, r, h, ß, 6, co 
h X 
C, B 
f> i 
23 letters in all. 
Zeuthen. 
n, a, 8, k, p, a 
h, q, k, t, 7 ) s 
c, r, h, ß ; m 
h X 
B, U,0 
f, i d, g, e 
27 letters in all. 
Here for Zeuthen’s k, h, I have written k, h, viz. these numbers represent the 
Pliickerian equivalents of the number of double points for the nodal and cuspidal curves 
respectively. Zeuthen considers also the general node, say (£ (/¿, v, y + y, z + £ u, v), 
see 416, this includes the cnicnode G and off-point <u, and accordingly he includes 
under it and takes no special notice of these singularities, but it does not properly 
include, and he takes special notice of, the binode B; it does not extend to the 
case where the tangent cone breaks up into cones each or any of them more than 
once repeated, and accordingly not to the case of a unode U where the tangent 
cone is a pair of coincident planes. He introduces this singularity, and also the 
singularity of the osculating point 0 which is understood rather more easily by means 
of the reciprocal singularity of the osculating plane O', this is a tangent plane 
meeting the surface in a curve having the point of contact for a triple point; and he 
disregards my unexplained singularity 6. The letters s, m do not denote singularities; 
s is the class of the envelope of the osculating planes of the nodal curve, m the