12 HENRY C. WENTE
This follows from the construction procedure indicated in (2).
4) There are in general six parameters, the constant a in (2.18),
the initial conditions a(u ), ft(u ), a'(u ), ft9 (u ) and co(u ,v ).
o o o o o o
Taking into account the translation of coordinates, we obtain a
four-parameter family of distinct solutions to (2.16).
We shall solve the system (2.18) by the method of separation
of variables applied to the Hamiltonian-Jacobi equation. We look
for a solution £(a,/?,h,k) to the P.D.E.
( 2 . 2 3 ) 6^ S = aaft - a2 ft2 - Aft2 - Ba2 + h
Make t h e c h a n g e of v a r i a b l e s
aft = s + t - 4-/AB
( 2 . 2 4 )
(VS
a -
VA
ft)2
= -
st .
With respect to these new variables the Hamilton-Jacobi equation
becomes
s ( s - ^-/AB)B 2 - t ( t - 4VAB)e 2 = g ( s ) - g ( t )
s t
( 2 . 2 5 )
g ( s ) = - s 3 + (a + 6-/AB)s2 + (h - 4a"/AB - 8AB)s + k .
Writing &(s,t) = P(s) + Q(t) separates the variables allowing us
to find B by quadratures. We differentiate with respect to the
parameters h, k setting 0, = u - u and 9, = k' where k' is a new
h o k
parameter. This determines s and t. In fact, one obtains the
following recipe. Let s(X), t(X) be distinct solutions to
s'(X)2 = s(s - 4yAB)g(s)
(2.26)
t'(X)2 = t(t - 4VAB) g(t)
where X = X(u) is determined by the condition
(2.27) X'(u) =
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