109 
122] A SURFACE OF THE SECOND ORDER INTO ITSELF. 
Next considering the equations connecting x, y, z, w with 77, £, co, we see that 
%i + 2/1 2 + z x 2 + w x 2 = ( £ + vrj — /j,£ + aco) 2 
+ (— + 77 + + bco) 2 
+ ( yi; — A.77 + £ + c co) 2 
+ (— a% — br) — c£+ o)) 2 . 
We are thus led to the discussion (in connexion with the question of the trans 
formation into itself of the form x 2 + y 2 + z 2 + w 2 ) of the new form 
( x + vy — fjtz + aw) 2 
+ (—vx+ y + \z + bw) 2 
+ ( fJLX—\y+ z + cw) 2 
+ (—ax — by — cz + w) 2 ; 
or, as it may also be written, 
(x 2 + y 2 + z 2 + vf) + {vy — /iz + aw) 2 + (\z — vx+ bw) 2 + (fix — \y + cw) 2 + (ax + by + cz) 2 . 
Represent for a moment the forms in question by U, V, and consider the surfaces 
U = 0, V = 0. If we form from this the surface V+qU— 0, and consider the dis 
criminant of the function on the left-hand side, then putting for shortness 
k = \ 2 + fi 2 + v 2 + a 2 + b 2 + c 2 , 
this discriminant is 
{(? + l) 2 + k (q + 1) + cf) 2 } 2 , 
which shows that the surfaces intersect in four lines. Suppose the discriminant vanishes; 
we have for the determination of q a quadratic equation, which may be written 
q 2 + (2 + k) q + K — 0; 
let the roots of this equation be q n q n \ then each of the functions q,U + V, q„U+ V 
will break up into linear factors, and we may write 
q,U+V = R / S / , 
q u U+V = R t fl„. 
(U and V are of course linear functions of Rfi, and R„S,j) forms which put in 
evidence the fact of the two surfaces intersecting in four lines. 
The equations 
x x + = 2£, yi+y 2 = %v> z 1 +z 2 = 2£, w x + w 2 = 2co,