[m‘poxazp] NUMBER OF REPRESENTATIONS OF A NUMBER 81 
The Twisted Biquadratic Curve of the First Species. 
The twisted biquadratic curve of the first species is defined as the 
complete intersection of two quadratic surfaces, excluding the case when 
such intersection consists of a straight line and a twisted cubic. Salmon, 
Geometry of Three Dimensions, p. 312. 
This curve admits of a parametric representation by means of elliptic 
functions. Appell and Lacour, Fonctions Elliptiques; Halphen, Traité 
des Fonctions Elliptiques, vol. II., p. 450. 
Let x,, x,, x, z,denote homogeneous co-ordinates. Then by a proper 
location of the tetrahedron of reference it is possible to express the points 
on the curve as follows: 2,:%,:%,:%,= p'u:p'u:pu:l, where pu 
is the fundamental elliptic function of Weierstrass. The following theo- 
rems are easily proved. See Halphen, loc. cit. : 
1. The necessary and sufficient condition that four points whose 
arguments are %,, U,, U3, U, may lie in one plane is 
u,+u, + us + u, = a period of p u. 
If we denote a primitive pair of periods of p u by 2 w, and 2 w; and 
w, + w; by w, we may write the above relation 
u, + uy + uz + u,=o (mod. 2 w,, 2 ws). 
2. Every quadric surface containing the curve is characterized by an 
argument + win such a way that the generating lines of one system of 
the surface meet the curve in two points, the sum of whose arguments is 
u, and of the other system in two points the sum of whose arguments is 
—u; and, moreover, to every argument + x corresponds a quadric 
surface containing the curve. 
If u = —u (mod. 2w,, 2w;) then u = 0, w,, w, Ws, and we get four 
surfaces whose two systems of generators coincide. These are the four 
cones on which the curve lies. 
Suppose now we consider any chord of the curve. We may repre- 
sent the sum of the arguments of the points where it meets the curve by 
—2u. Then the other points where any plane through this chord 
mects the curve are given by a relation 
—2ut+tu,+ u,=o (mod. 2w,, 2w;). 
Suppose the two points w,, wu, coincide at u. Then—Su+ 2u =o, 
or 2u,=2u,oru,=u,u+ w,u + wu + wz, or, in other words, 
through any chord pass four tangent planes. 
Theorem. 
3. The cross-ratio of the four tangent planes which pass through any 
chord is the same for all chords. Hence this cross-ratio, which is un- 
altered by any collineation, may be regarded as an absolute invariant of 
