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ON A SINGULARITY OF SURFACES.
[402
Section by the non-special plane through the tangent line, viz. the plane y = 0.
The equation is
(a, b, c, /, g, h\-z 2 , - x, xz)- = 0,
or what is the same thing,
bx 2 — 2fx 2 z + 2 hxz 2 + coc-z 2 — 2yxz 3 + az 4 = 0.
Writing as usual x = Az^ + &c. we have
g = 2, bA~ + 2 hA + a = 0,
and since ab — h 2 is by hypothesis not = 0, A has two unequal values; we have at
the origin two branches x = A^z 2 4- B x z 3 + &c., x = AS- + B. 2 z 3 + &c., having the common
tangent x = 0 (viz. this is the tangent x = 0, y = 0 of the nodal curve), and with a
two-pointic intersection of the two branches, that is, the point at the origin is an
ordinary tacnode.
Section by the osculating plane x — 0.
The equation is
(a, b, c, f g, h\y - z 2 , yz, - y 2 ) 2 = 0.
We may write y = z 2 + Az* + &c., we at once find g = 3, and then
(a, b, c, f g, h'$Az 3 +&c., z 3 + 8zc., — z 4 + &c.) 2 = 0,
that is
(a, h, b^A, 1) 2 = 0.
A has two unequal values, and the branches through the origin are
y = z 2 + A^z 3 + B^z* + &c., y = z 2 + A .,z 3 + B.^z 4 + &c. ...,
viz. the branches have the common tangent line ?/ = 0 (the tangent x = 0, y = 0 of
the nodal curve), but in the present case a three-pointic intersection.
Section by one of the tangent planes (a, h, b\y, — x) 2 = 0.
Writing y — —mx, and therefore (a, h, b\m, — 1) 2 =0, the equation is
(a, b, c, f g, mx — z 2 , — x — mzx, xz — m 2 x?) 2 = 0,
which represents of course the projection of the section on the plane z = 0, x = 0, but
which (since there is no alteration in the singularities) may be considered as
representing the section itself. Developing, the coefficient of x 2 is am 2 -1- 2hin + b, which
is = 0, and the equation becomes
2m 2 (gm +f) x 3 + cm 4 x 4
+ 2 [hm 2 + (b — g)m —/} x 2 z + 2m 2 (fm — c) x*z
+ 2 (am + h) xz 2 + (b + 2g) m 2 — 2fm + c x 2 z 2
+ 2 (hm — g) xz 3
+ a z* — 0,
so that the curve has at the origin a triple point, the tangent to one branch being
the line x — 0 (the tangent x = 0, y = 0 of the nodal curve).
