435] 
ON THE SIX COORDINATES OF A LINE. 
79 
Case of the fivefold relation. 
36. The fivefold relation 
a , b, c , 
/ > 9 > A 
dry , b] ? Cq ; 
A, 9i, h x 
expresses that the quantities (a, b, c, f g, h) are proportional to (cq, b u c lt f, g lt hfi 
As the former set are by hypothesis the coordinates of a line, the given set (cq, b 1} cq, f 1} g lt hfi) 
must, it is clear, also be the coordinates of a line, and the relation then expresses 
that the line (a, b, c, f g, h) coincides with the given line. 
Case of the fourfold relation. 
37. The fourfold relation is 
a , b , c , 
f> 
9 , 
h 
= 0, 
oq > bi, Ci, 
À, 
9i, 
K 
a 2 , b 2 , c 2 , 
u 
92, 
h 2 
or what is the same thing, we have the six equations Aa + Ajcqq- A 2 a 2 = 0, &c., involving 
the indeterminate quantities A, A 1? A 2 . If the coefficients 
(cq, bn c 1; f, gi, /q), (a 2 , b 2) c 2 , f, g 2 , A 2 ) 
are not either set the coordinates of a line; then substituting the foregoing values 
— Aa = Ajcq + A 2 cq, &c. in the equation af+bg + ch— 0, we have a quadric equation in 
(A x : A 2 ) : and for each root of this equation, the coefficients A 1 a 1 +A 2 cq, &c. will be the 
coordinates of a line. There are thus in general two derived lines; and the fourfold 
relation expresses that the line (a, b, c, f g, h) coincides with one or other of these 
derived lines. There is no real difference if one or the other of the two sets 
(cq, bn Cnji, gi, Ai), (a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ), or if each set, are the coordinates of a line; 
one of the derived lines or both of them will in these cases coincide with one or 
both of the given lines. And if the quadric equation has equal roots, then instead of 
two derived lines there is a twofold derived line, and the line (a, b, c, f g, A) must 
coincide with this twofold line. 
38. A case presenting peculiarity is however that in which the coefficients of the 
quadric equation vanish identically; this is only so when the coefficients (a ly b lf <h>fi, g x , h x ) 
and (a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ) are the coordinates of two intersecting lines. The equations 
— Aa = Ajoq + A 2 a 2 , &c. here show that every line whatever which meets each of the 
two lines (cq, b 1} c 1 , f, g x , /q) and (a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ) meets also the line (a, b, c,f g, h); 
that is, the line (a, b, c, f g, h) is any line whatever in the plane and thiough the 
point of intersection of the two intersecting lines. We see moreover that not only 
af 2 + b x g 2 + c x h 2 4- a 2 f + b 2 g x + c 2 /q = 0,