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,