82 ON THE SUPERLINES OF A QUADRIC SURFACE [570 
other, according as the sign is + or —. It is more convenient to take the determ 
inant to be always + , and to write the equations in the form 
u = k (ax 4- (3y +7z ), 
v = k (a'x + (3’y 4- 7 ’z ), 
w = k (cl'x + ¡3"y 4- 7"z), 
where k=± 1, and the superline is of the one or the other kind, according as the 
sign is + or —. 
Now considering two superlines, we may write 
u = ax + fry 4- yz , u = — k (ax + by + cz ), 
v = a'x 4- ¡3'y +7'z , v = — k (a'x + b'y 4- c'z ), 
w = a'x 4- (3"y + 7"z, w = — k (a"x + b"y + c"z). 
If the superlines intersect, then 
(a + ka ) x + (/3 -1-kb ) y + (7 + kc ) z = 0, 
(a' + ka')x + (/3' + kb') y + (y + kc') z = 0, 
(a" + ka") x + (/3" + kb") y + (7" 4- kc") z = 0, 
viz. the determinant formed with these coefficients must be =0. The condition is at 
once reduced to 
1 + k 3 + (k + k 2 ) (aa + b/3 + cy + a'a + b'/3' + c'y + a!’a!' + b"/3" + c"y”) = 0, 
viz. it is satisfied when & = — l, that is, when the superlines are of the same kind; 
but not in general when & = +l. 
If k = + 1 the condition will be satisfied if 
1 + aa + b/3 + c 7 + a a! + b'f3' + c'y' + a"a" + b"/3" + c"y" = 0, 
and it is to be shown that then the three equations reduce themselves not to two 
equations, but to a single equation. 
It is allowable to take the second set of equations to be simply u = — kx, v = — ky, 
w = — kz\ for this comes to replacing the analytically rectangular system ax + by + cz, 
a'x + b'y + c'z, a"x + b"y + c"z by x, y, z. Writing also & = + l, the theorem to be proved 
is that the equations 
(a + l)x + (3y + yz = 0, 
a’x +(& + l)y + y'z = 0, 
a"x + (3"y + 7 "z — 0, 
reduce themselves to a single equation, provided only 1 + a + 0' + y" = 0; or, what is 
the same thing, we have to prove that the expressions /3" — y, 7 — a", a' — ¡3 each 
vanish, provided only 1 + a + /3' + y" = 0. This is a known theorem depending on the