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,—