740] 
ON CERTAIN ALGEBRAICAL IDENTITIES. 
131 
In the case of the elliptic functions, we may consider the quadriquadric curve 
y 2 = 1 — x 2 , z 2 = 1 — k 2 x 2 , 
so that the coordinates of a point on the curve are sn u, cn u, dn u, Taking then 
P 0 , P 1 , P 2 , points on the curve, and (x 0 , y 0 , z 0 ), (x x , y l} z x ), (x 2) y 2 , z. 2 ), the coordinates of 
these points respectively, we have in the same way, from u 2 — u 0 = (n 2 — %) + (tq — u 0 ), 
three equations, of which the first is 
(1 - kWx 2 2 ) {x 2 y x z x - x x y 2 z 2 ) (y 0 y 1 + XqZqXjZj^) (z 0 z, + tex^x^) 
x&oZo - X 0 y 2 z, _ + (1 - k 2 xoW) (x^qZ, - x 0 y x z x ) (y x y 2 + x 1 z 1 x&) (z x z 2 + k 2 x x y x x#)Z) 
1 — k 2 x 0 2 x 2 2 (1 — k 2 x 0 2 xi 2 ) 2 (1 — k 2 x x 2 x 2 2 ) 2 — k 2 (xfloZo — x 0 y x z x f {oc 2 y x z x — x x y 2 z<Zf' 
The form of the right-hand side is 
A 4- Bx l y x z l 
C + Dx 1 y 1 z 1 ’ 
where A, B, G, D are each of them rational as regards x{-; and it is easy to see 
that the equation can only subsist under the condition that we have separately 
x 2 y^ x 0 y 2 z 2 A B 
1 — k 2 x 0 2 x 2 C D ’ 
implying of course the identity AD — BG=0. The values of B and D are found 
without difficulty; we, in fact, have 
B = 2k 2 (x.f - x 0 2 ) (xPy^y.z, + x 0 x 2 y x 2 z x 2 ), 
D = 21c- (x 2 y 0 z 0 + x 0 y 2 z 2 ) (x?y^y 2 z 2 + x 0 x 2 y x 2 z x 2 ), 
so that, comparing the left-hand side with B h- D, we have the identity 
^2/oV - ®*yZzi — (x 2 2 - x 0 2 ) (1 - k 2 xZxo 2 ), 
which is right. The comparison with A G would be somewhat more difficult to effect.