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 -
ft)2 = - st
(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
a + 6-/AB
P = 3
I = 16AB + 4h + -• a2
AV\ j——
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
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