114 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [123 
and, this being so, p and q are the parameters by which the point in question is 
determined. We may for shortness speak of the surface 
k (x 2 + y 2 + z 2 + w 2 ) + ax- + by 1 + cz 2 + div 2 = 0 
as the surface (k). It is clear that we shall then have to speak of 
x 2 + y 2 + z 2 + w 2 = 0 
as the surface (oo ). 
I consider now a chord of the surface (oo) touching the two surfaces (k) and 
(,k'); and I take 0, $ as the parameters of the one extremity of this chord; {p, q) 
as the parameters of the point of contact with the surface (k); p', q' as the parameters 
of the point of contact with the surface (k'); and 6', as the parameters of the 
other extremity of the chord; the points in question may therefore be distinguished 
as the points (oo ; 9, <£), (&; p, q), (k'; p', q'), and (oo; 0', <//). The coordinates of the 
point (oo ; 9, cf>) are given by 
x : y : z : w= V(a + 0) (a + <j>) -r V(a — b) (a — c) (a — cl) 
: V(6 + 0) (b + <f>) 4- V(6 — c) (b — d) (b — a) 
: */(c + 0) (c + <f>) + *J(c — d) (c — a) (c — b) 
: V(ci + 0) (d + (f>) -p \/{d — a) (d — b)(d — c); 
those of the point (k; p, q) by 
x : y : z : w— V(a + p) (a + q) -4- V(a — b) (a — c) (a — d) Va + k 
: V(o + p) (b + q) 4- V(6 — c)(b — d) (b — a) *Jb + k 
: V(c + p) (c + q) -p V(c — d) (c — a) (c — b) Vc + k 
: '/(d + p) (d + q)\/(d — a) (d — b) (d — c) \bd + k 
and similarly for the other two points. 
Consider, in the first place, the chord in question as a tangent to the two 
surfaces (k) and (kIt is clear that the tangent plane to the surface (k) at the 
point (k; p, q) must contain the point (k'; p, q ), and vice versa. Take for a moment 
q, f, co as the coordinates of the point (k; p, q), the equation of the tangent 
plane to (k) at this point is 
X (a + k) %x = 0 ; 
or substituting for f,... their values 
X (x V(a + p) (a + q) Va + k V(a — b) (a — c) (a — d)) = 0 ;