843] 
WITH THE THEORY OF ELLIPTIC FUNCTIONS. 
323 
41—2 
which is a surface having the line {a', b', c', f, g', li) for a generating line. To 
verify this, observe that for the line in question we have 
h'y — g'z + a'w = 0, 
— h'x . + f'z + b'w = 0, 
g'x + f'y . + c'w = 0, 
— a'x — b'y — c'z . = 0, 
equivalent of course to two independent equations: these give h'x = f'z + b'w, 
h'y = g'z — aw, values which substituted in the quadric equation satisfy it identically. 
And for the last-mentioned quadric surface we have the theorem that, if {a, b, c, f, g, h) 
are the coordinates of a generating line, then 
a a' 
f =± f” 
b_ V c__ c 
9 V’ h ± h' 
where obviously the sign + belongs to a generating line of the same system with 
the line (a, b', c', /', g', h'), and the sign — to a line of the other system. 
Taking the sign —, we thus see that if (a, b, c, f, g, h), (a', b', c', f, g, h') are 
the coordinates of lines of the two systems respectively, then 
af + af = 0, bg' + b'g = 0, ch' + c'h = 0 ; 
where observe that the resulting equation 
af' + af + bg' + b'g + cli + c'h = 0, 
is the condition which expresses that the two lines meet each other. 
Consider now the quadriquadric curve 
U l = Ax 3 + By 1 + Gz 2 + Dw 2 = 0, 
Ui = A'x 2 + B'y 1 + G'z 2 + B'w 2 = 0; 
and let (a, b, c, f g, h) and {a', b', c, f, g', h') be the coordinates of two lines meeting 
each other, and each meeting the quadric curve twice: or, again, let these be lines 
joining in pairs the four intersections of the curve by an arbitrary plane: or, again, 
let them be the nodal lines of the binodal quartic cone having an arbitrary vertex 
and passing through the curve. The two lines are generating lines, belonging to the 
two systems respectively, of a properly determined quadric surface U + A, U' = 0 passing 
through the curve: and by what precedes, we have 
af' + af = 0, bg' -t- b'g = 0, ch' + c'h = 0, 
the fundamental theorem which I wished to establish. 
Writing in the equations w=l, we have in particular the quadriquadric curve 
y 2 = 1 — x 2 , z 2 = 1 — k 2 x 2 , equations which are satisfied by x = sn u, y = cn u, z = dn u. 
Consider on the curve four points belonging to the arguments u x , u 2 , u 3 , u A respectively;