294 
[834 
834. 
ON THE ADDITION OF THE ELLIPTIC FUNCTIONS. 
[From the Messenger of Mathematics, vol. xiv. (1885), pp. 56—61.] 
Mr Forsyth’s Note [l.c., p. 23] on my “Formula in Elliptic Functions” has supplied 
a missing link, and I am now able to obtain the addition formulae very simply 
from the application of Abel’s theorem to the Quadriquadric Curve. 
I remark that, instead of coplanar points 1, 2, 3, 4, it is advantageous to consider 
coresidual points 1, 2 and 3, 4; that is, pairs 1, 2 and 3, 4, which are each of 
them coplanar with one and the same pair of points 5, 6. The difference is as follows : 
for the coplanar points 1, 2, 3, 4, we have 
giving 
du Y + dm, + du 3 + du 4 = 0, 
u 1 + u 2 + u 3 + u 4 = C, 
and for the addition theory it is necessary to have (7 = 0; for the coresidual points, 
we have 
u x + u 2 + u s +u 6 = C, u 3 +u 4 + u ri + u 6 = G; 
and thence u x + u 2 = u 3 + u A , irrespectively of the value of C. 
As to the general theory of a curve in space, observe that v when this is a complete 
intersection of two surfaces 
f{oc, y, z, w) = 0, g(x, y, z, w) = 0, 
then at the point (x, y, z, w), if 
(x + dx, y + dy, z + dz, w + dw) 
are the coordinates of the consecutive point, the six coordinates of the tangent line 
y dz — z dy, zdx — x dz, xdy — y dx, x dw — w dx, y dw — w dy, z dw — w dz. 
are