116 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [123
a set of equations which may be represented by the single equation
yfr (x +p) (x + q) (x +p') (x + q') — (px + cr) 2 (x + k)(x + k') = ^(x — a)(x — b)(x — c) (x — d),
where x is arbitrary; or what is the same thing, writing — x instead of x,
X + a) (x + b) (x + c)(x + d) + (px — cr) 2 (x — k)(x — k') = yjr(x—p)(x — q) (x —p') (x — q').
Hence, putting
dx
*J(x + a) (x + b) (x + c) (x + d) (x — k) (x — k') ’
we see that the algebraical equations between p, q; p\ q' are equivalent to the
transcendental equations
Yip + nq ± Up' ± n^' = const.
Yi t p ± + II t p' ± I\- t q' = const.
The algebraical equations which connect 9, (f> with p, q; p', q', may be exhibited
under several different forms; thus, for instance, considering the point (oo ; 9, cf>) as
a point in the line joining (&; p, q) and (k'; p', q'), we must have
V(a + p) (a + q) -f Va + k, V(6 +p) (b + q) -r- \/b + k,...
V(a +p') (a + q')-i-\/a + k', V(b +p') (b + q')+ \!b + k'
V(a + 9) (a + </>), V(6 + 9) (b + 4>)
i.e. the determinants formed by selecting any three of the four columns must vanish;
the equations so obtained are equivalent (as they should be) to two independent
equations.
Or, again, by considering (oo ; 9, (/>) first as a point in the tangent plane at
(k; p, q) to the surface (k), and then as a point in the tangent plane at (k'; p', q')
to the surface (k'), we obtain
2 ( (b - c) (c — d) (d — b) V(a + p) (a + q) V(a + k) V(a + 9) (a + <£)) = 0,
2 ( (b — c) (c — d) (d — b) V(a +p') (a + q') V(a 4- k') \/(a + 9) (a f <£)) = 0.
Or, again, we may consider the line joining (go ; 9, cf>) and (k; p, q) or (k'; p', q'),
as touching the surfaces (k) and (k'); the formulae for this purpose are readily
obtained by means of the lemma,—