123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. 117 
“The condition in order that the line joining the points (£, y, £, co) and (f', y, &/) 
may touch the surface 
sue 2 + b y 2 + c z 2 + d w 2 = 0 
is 
Sab {fy - %y) 2 = 0, 
the summation extending to the binary combinations of a, b, c, d.” 
But none of all these formulae appear readily to conduct to the transcendental 
equations connecting 6, <f> with p, q; p', q'. Reasoning from analogy, it would seem 
that there exist transcendental equations 
+ lid + IIcf) + FTp ± Up' = const. 
± 11,0 ± II,6 ± Ii,p + Ii,p' = const., 
or the similar equations containing q, q', instead of p, p', into which these are changed 
by means of the transcendental equations between p, q, p', q'. If in these equations 
we write 6', (f)' instead of 6, c/>, it would appear that the functions lip, lip', II,p, II,p’ 
may be eliminated, and that we should obtain equations such as 
+ 110 + Tift ± n0' + Hep' = const. 
+ 11,6 ± U,(f> + FT,6' + II,cj)' = const. 
to express the relations that must exist between the parameters 6, cf> and 6', <f)' of 
the extremities of a chord of the surface 
oc 2 + y 2 + z 2 + w 2 = 0, 
in order that this chord may touch the two surfaces 
k {pc 2 + y 2 + z 2 + w 2 ) + ax 2 + by 2 + cz 2 + dw 2 = 0, 
k' {x 2 + y 2 + z 2 + w 2 ) + ax 2 + by 2 + cz 2 + div 2 = 0. 
The quantities k, k', it will be noticed, enter into the radical of the integrals 
Ilx, Ii,x. This is a very striking difference between the present theory and the 
analogous theory relating to conics, and leads, I think, to the inference that the theory 
of the polygon inscribed in a conic, and the sides of which touch conics intersecting 
the conic in the same four points, cannot be extended to surfaces in such manner as 
one might be led to suppose from the extension to surfaces of the much simpler 
theory of the polygon inscribed in a conic, and the sides of which touch conics having 
double contact with the conic. (See my paper “ On the Homographic Transformation 
of a surface of the second order into itself,” [122]). 
The preceding investigations are obviously very incomplete; but the connexion 
which they point out between the geometrical question and the Abelian integral 
involving the root of a function of the sixth order, may I think be of service in 
the theory of these integrals.