834] 
ON THE ADDITION OF THE ELLIPTIC FUNCTIONS. 
295 
But considering the line as the intersection of the two tangent planes 
and 
df v df df „ df 
J-X + -fY+±Z+-/- 
dx dy dz dw 
coordinates are 
d 9 y dg v dg dg 
dx + dy + dz + dw 
d(f g) 
d(x, w) ’ 
d (f g) d (f g) d (/, g) 
d (y, tu) ’ d(z, w)’ d {y, z) ’ 
the six quotients 
(yd, *dy)\ d ( £ wy 
d(f, g) 
d(x, y)’ 
are equal to each other, and may be put = dco. 
Considering any two quadric surfaces, there is in general a system of four con 
jugate points, or points such that in regard to each of the quadrics the polar plane 
of any one of the points is the plane through the other three points. And then 
taking x — 0, y — 0, z — 0, tv = 0 for the equations of the faces of the tetrahedron 
formed by the four points, the equations of the quadric surfaces will be of the form 
a« 2 + b y* + cz 2 + d w 2 = 0, 
aV + h'y- -f cV + d'w 2 = 0 ; 
we then have the six quotients 
{y dz — z dy)/(ad' — a'd) xw, &c., 
equal to each other, and each = dco. Here dw is homogeneous of the degree zero 
in the coordinates (x, y, z, tv), or, what is the same thing, it is a differential 
F ^ d — , say it is = du ; and taking the integrals always from one and the same 
fixed point on the curve, we have each point of the curve corresponding to a determ 
inate value of a parameter u. 
Supposing that u r , w 2 , u 5 , u 6 are the values of u, belonging to any four coplanar 
points 1, 2, 5, 6; then, by Abel’s theorem, diiy + du 2 + du 5 + du 6 = 0; that is, we have 
u r + u 2 + u B + u 6 = C, 
as the condition in oi’der that the four points 1, 2, 5, 6 may be coplanar; similarly, 
we have 
tt-j -j- W-4 4 u B -|- Uq = (J, 
as the condition in order that the four points 3, 4, 5, 6 may be coplanar ; and we 
have therefore 
tly -f- u 2 = ti 3 + u 4 , 
as the condition that the two pairs of points 1, 2 and 3, 4 may be coresidual.