123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. 115 
or taking for x,... the coordinates of the point (k', p', q'), we have for the conditions 
that this point may lie in the tangent plane in question, 
2 (V (a +p) (a + q) V (a +p') (a + q') V(a + k) -r V(a + k') (a — b) (a — c) (a — d)) = 0 ; 
or under a somewhat more convenient form we have 
2 ((b — c) (c — d) (d — b) V(a + p) (a + q) V(a + p') (a + o') - = 0, 
v V a + &v 
for the condition in order that the point (&', p', 5') may lie in the tangent plane at 
(k; p, q) to the surface (k). Similarly, we have 
2 ((b — c)(c — d) (d — b)^(a + p) (a + q) V(a + p')(a + q') ^- + \ \ = 0, 
\ Wa + k/ 
for the condition in order that the point (k, p, q) may lie in the tangent plane at 
(k'; p', q') to the surface (k'). The former of these two equations is equivalent to 
the system of equations 
V(a +p) (a + q) (a + p') (a + q') = ^ + pa + va\ 
and the latter to the system of equations 
V(a + p) (a + q) (a + p') (a + q). 
ci ~t~ k 
ci + k 
A -f- fx ci + v 
where in each system a is to be successively replaced by b, c, d, and where A, p, v 
and A', p, v are indeterminate. Now dividing each equation of the one system by 
the corresponding equation in the other system, we see that the equation 
x + k A + px + vx 2 
x + k' A' + px + v'x 2 
is satisfied by the values a, b, c, d of a; and, therefore, since the equation in x is 
only of the third order, that the equation in question must be identically true. We 
may therefore write 
A + px + vx 2 = (px + a)(x + k), A' + px + v'x 2 = (px + a)(x + k'), 
and the two systems of equations become therefore equivalent to the single system, 
V(a +p) (a + q) (a +p') (a + q) = (pa + a) V(a + k) (a + k'), 
V(6 +p)(b + q)(b + p') (b + c{) = (pb + cr) V(b + k)(b + k r ), 
V(c +p) (c + q) (c +p') (c + q) = (pc + a) V\c + k) (c + k'\ 
V(d+p)(d + q) (d +p') (d + q') = (pd + a) \/(d + k) (d + k'), 
15—2