Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 345
to any one of the other types. But in order that two such quartics with given coefficients should be transformable into one another it would of course be necessary that
there should be a relation between their coefficients equivalent to the condition that
the corresponding cubics should have their absolute invariants equal.
It should be noted moreover that we have supposed our quartic surfaces to be
the most general of their respective types. For special relations between the coefficients
one of the quartics might become a cone-a case that we have excluded-or the corresponding cubic cone might become rational or degenerate, in which cases no integrals
of the first kind could exist.
~ 6. NUMERICAL GENUS OF SURFACES WHICH ADMIT OF INTEGRALS OF THE FIRST K1ND.
It appears from the preceding analysis that the only quartic surfaces which admit
of integrals of the first kind are cones or birational transformations of cones; consequently the (numerical) genus* is in each case negative; the numbers being -3 for
a non-singular quartic cone, - 2 for a quartic cone with one double line, and otherwise -1.
In the course of an investigation dealing with quintic surfaces I have miet with
several surfaces which admit of integrals of the first kind, and these surfaces likewise
have negative genus. On the other hand Humbert in his well-known memoir on hyperelliptic surfaces has given some octavic surfaces which admit of integrals of the first
kind but are of positive genus. Whether such integrals can exist on any surface of
order 5, 6, or 7 with positive genus appears to be at present unknown.
~ 7. GEOMETRICAL CHARACTERISTICS OF THE FIVE SURFACES.
The surface (A) occurs in Kummer's well-known paper on quartic surfaces which
contain families 'of conics+. The surface touches itself at each of the points, y=z=w=O0,
x=z=w=O; any plane section through these points consists therefore of a plane quartic
curve touching itself twice, that is of a pair of conics having double contact. The
two points belong to a class of singular points of surfaces which seem to have been
little studied; such a point may be defined as a uniplanar double point, which is further
a quadruple point on the section by the tangent plane, and is consequently a tacnode
on a general section through the point. Kummer speaks of a " Selbstberihrungspunct";
tacnode or tacnodal point seems a convenient English name~. It can easily be seen
that a tacnode diminishes the order of the reciprocal surface by 12, so that for this
purpose it is equivalent to six ordinary double points. As Picard and Simart point out,
* Genre numérique, deficiency. Cf. Cayley's paper t Liouville, sér. iv. t. 9 (1893).
"On the deficiency of certain surfaces," quoted before; $ Crelle, t. 64 (1864).
Picard et Sirmart, ch. viII. ~ iv; Castelnuovo & Enriques: ~ According to Picard and Simart this is the name
" Sur quelques récents résultats dans la théorie des surfaces given by 'les géomètres anglais,' but I have not been able
algébriques," Math. Ann. t. XLVIII. (1897). to find any such authority for it.
VOL. XVIII. 44