ENNEPER TYPE IMMERSIONS 11
where p(u,X) = - (4A + a2)X4 - 4a'X3 + 6^X2 + 4/?'X - (4B + ft2).
Here p(u,X) is a fourth degree polynomial in X with the
coefficients functions of u. It is necessary that p(u,X) be
CO
positive for some positive values of X = e in order for a
solution co(u,v) to exist.
A fourth degree polynomial has two invariants I and J defined
as follows. If
/ v * 3 2
p ( x ) = a x + 4 a x + 6 a x +4a x + a
O 1 2 3 4
( 2 . 2 1 ) I = a a
o 4
4a a + 3a
1 3 2
d e t
a a a
o
I
2
a a a
1 2 3
a a
2 3
while the discriminant A = I - 27J . A direct calculation
shows that the invariants associated with the polynomials p(u,X)
in (2.20) are constant and independent of u (i.e. they are
integrals of the system (2.18)).
2) Suppose that for u = u we have p(u ,X) in (2.20) positive
o o
for some X 0. We may use (2.20) to find co(u , v) and then
o o
(2.17) to find co(u,v). We observe that (2.17) is a Ricatti
CO
equation in X = e .
(2.22)
2X a(u)X +
ft(u)
3) If co'(u ,B) = 0 then co(u,v) is symmetric about the line v = B
v o
satisfying
c o (u,B) = 0
v
co(u,B + v) = co(u,B - v).
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