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).