435] 
ON THE SIX COORDINATES OF A LINE. 
83 
where (a x , ...), (a 2 (a 3 ,(a 4 ,...), (a g ,...), are each the coordinates of points in a line. 
The preceding mode of dealing with the question is inapplicable, since there is not in 
general any line which meets the five given lines; in the particular case, however, 
where the five given lines are met by a single line, say when they have a common 
tractor, then the line (a, b, c, f g, h) is any line meeting this common tractor. The 
general case is that of the involution of six lines, mentioned No. 25, and the con 
sideration of which was deferred. 
47. The onefold relation implies that we can find multipliers X, g, v, p, a, r, such 
that 
X<z + gb + vc + pf + erg + rh = 0, 
X«! + gb x + vc x + pf x + og x + rh x = 0, 
Xa s + gb 5 + vc 5 + pf5 + a [/s + t K = 0, 
we may by means of the last five equations determine the ratios of X, g, v, p, a, r, 
viz. these quantities will be proportional to the determinants formed out of the matrix 
b x , Ci, f, g x , h x 
ci-2, b 2 , c 2 , / 2 , g%, h 2 
a 3 , K c 3 , f 3 , g 3 , h 3 
b 4 , c 4 , g4, h 4 
a 5> b 5) c 5) f 5 , g B , h s 
and the first equation is then a linear relation in (a, b, c, f g ; It), expressing the 
relation that exists between these coordinates. 
48. Consider an arbitrary point 0 on the line (a, b, c, f, g, h); taking this point 
as origin, the coordinates of 0 are 0, 0, 0, 1; and if x, y, z, w, are the coordinates 
of any other point on the line, then writing 
x, y, z, w, 
0, 0, 0, 1, 
we find 
a : b : c : f : g : h = 0 : 0 : 0 : x : y : z ; 
\a + gb + vc + pf 4- o-g + tIi = 0 
and the equation 
becomes simply px + cry + tz = 0 ; viz. this equation expresses that the line (ci, b, c,f g, h), 
assumed to pass through a given point 0, lies in a determinate plane £1 through this 
point. 
49. To construct this nlane O, I consider any four of the five given lines, say