82 ROYAL SOCIETY OF CANADA 
the curve. It will also appear as an absolute invariant of the elliptic 
functions which give the parametric representation of the curve. 
It is the object of this note to give a direct proof of this theorem, 
which Halphen proves by reference to a developed theory of the plane 
non-singular cubic. 
Proof. 
By theorem 2 the tangents to the curve at u, u+ w,, w+ w,,u + Ws, 
lie on a quadric surface containing the curve and also the chord, the sum 
of whose arguments was — 2 u. For these tangents meet the curve in 
ponte the sums of whose arguments are congruent to 2 u, modulo 2 w, 
2 Ws. 
By a known theorem of the geometry of quadric surfaces, the cross- 
ratio of the four planes by which any four generators of one system are 
projected from any generator of the other system, is independent of this 
generator. In calculating, therefore, the cross-ratio we wish, we may use 
any chord whose arguments have —2 u for sum. 
Take the chord joining the points 0, — 2u. Denote 0, w,, w,, ws by Wy, 
viz., put. 0 = w, and let A = 0, 1, 2, 3. 
Then the equations of the four planes are comprised in the expres- 
sion 
à a Ls aie Le La Œ, 
1 0 0 (LATE p'(—2u)p'( —2u) 1 
p'C—2u) p'(—2u) pC —2u) 1) or | vp’ (u+w))p(utw)t 
p' (u-tw,) p' (uw) p (uw) 1 
If we take a section of this pencil of planes by the plane x, = 0, we 
get a pencil of lines the cross-ratio of which is the same as that of the 
planes. We have then to find the cross-ratio of the four lines. 
a, [p Qu) —p Cu + w)] + x Cp Qu) + pu + w)] = 0 
or of the four quantities 
p’ (2u) +p’ (w+ wy) 
p Qu) —p (+ wy) 
The above expression may be transformed. Referring to Schwarz- 
Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen 
Functionen, p. 14, we have 
=0 
p'(w) + »’ 2 
pœ@rn=r [RERO] re -re 
putting w —=2u veut wy, 
p' Qu)+ p'(u + wy)—? 
p (2u)— p (u + w)) 

p (u —w,)=¢ — p (Qu) — p (u + %)). 
