414]
ON POLYZOMAL CURVES.
487
Now, from the two equations as they stand we can pass back to the two tetrazomal
equations, and the first-mentioned 2r 2 points are thus points of intersection of the
two tetrazomal curves—from the two equations after such reversal of the sign of
y/T, we cannot pass back to the two tetrazomal equations, and the last-mentioned 2r 2
points are thus not points of intersection of the two tetrazomal curves. The number
of intersections of the two curves is thus 8 x 2r 2 , = 16r 2 , as it should be.
Article Nos. 42 to 45. The Theorem of the Decomposition of a Tetrazomal Curve.
42. I consider the tetrazomal curve—
flU + VmF+ ffW + ^JpT = 0,
where the zomal curves are in involution,—that is, where we have an identical relation,
aU + bV + cW + dT = 0 ;
and I proceed to show that if l, m, n, p satisfy the relation
m n p
l
—h
a
n
—h
C
= 0,
b c d
the curve breaks up into two trizomals. In fact, writing the equation under the form
(y/lU + fmV + fnW) 2 — pT — 0,
and substituting for T its value, in terms of U, V, W, this is
(¿d 4-pa) TJ + (md + pb) V + (Ad +pc) W
+ 2 fmnd V VW + 2 fnld V WU + 2 flmd f TJV = 0 ;
or, considering the left-hand side as a quadric function of (V U, fV, V W), the condition
for its breaking up into factors is
= 0,
that is
or finally, the condition is
¿d + pa,
d flm,
d fin
d fml,
md -t-pb,
d fmn
d y/nl ,
d V nm,
wd + pc
p 2 (¿bed + meda + ndab + joabc) = 0,
l
—h
a
m
b +
n
—b
c
1 = 0-
43. Multiplying by ld+pa, and observing that in virtue of the relation we have
(¿d + pa) (md + pb) = ¿md 2 — pn,
(¿d + pa) (nd +pc ) = Ind 2 —r~ pwi,