ENNEPER TYPE IMMERSIONS 13

In theory we have found s and t as functions of u giving us the

solutions (a(u),/?(u) } . This is the classical procedure of Jacobi.

Suppose that A 0 and B is positive. In this case the

transformation (2.24) is real and we find

(-/ B

a -

VA

ft)2 = - st

(2.28)

(VB a + VA ft)2 = -(s - 4VAB)(t - 4-/AB) .

In the transformation either s 4-/AB and t 0 or conversely.

Therefore in solving (2.26) we look for solutions {s(\), t(\)}

satisfying s 4-/AB and t 0.

There is an ambiguity in recovering a,ft from s,t. If

ot(u), ft(u) satisfies (2.24) then so does -ot(u), -/?(u). The

proper choice becomes important in (2.20). It is necessary

to choose ot, ft so that the polynomial p(u, X)is positive for some

positive X. If p(u,X) is the polynomial corresponding to a,ft

then p(u,-X) is the polynomial corresponding to -a, -/?.

One may start with the polynomial g(s) given by (2.25) where

it is necessary that g(s) be positive for some s 4-/AB. Write

g(s) in the form

4 g(s) = 4(-s + p) - I(-s +p) + J

(2.29)

a + 6-/AB

P = 3

Then

I = 16AB + 4h + -• a2

(2.30)

AV\ j——

ft

o ^

*?

J = 4k + •—(a + 6-/AB) + •— a aAB

where these are the invariants of the polynomial p(u, X) appearing

in (2.20). In particular one finds that