86 
ON THE SIX COORDINATES OF A LINE. 
[435 
then if (A, B, G, F, G, H) be the coordinates of a tractor of these lines, we have 
(F, G, H, A, B, 0$a, b, c,f,g, h) = 0, 
(F, G, H, A, B, CT&ck, K *,/1, <7i, M = 
(F, G, H, A, B, G\a 3 , b ± , c 2 , /„ g 2 , K) = 0, 
(F, G, H, A, B, G\a 3 , b 3 , c 3 , f 3> g 3 , h 3 ) = 0. 
In virtue of these relations the ratios A : B : C : F : G : H are given linear 
functions of any one of these ratios or of an arbitrary ratio u : v; and we then have 
AF -f BG + CH — 0, a quadric equation for determining the unknown ratio. In the 
case of a twofold tractor, this equation must have equal roots; whence employing as 
usual the method of indeterminate multipliers, we find 
A Ait 4~ Ajfti 4~ A 2 ft 2 4- Ay^ = 0, 
B 4~ Ab 4" A4~ A 2 6 2 4" A 3 b 3 = 0, 
G 4" Ac 4" A^Ci 4“ A 2 c 2 4- A 3 c 3 = 0, 
F + \f 4- Axfx + A 2 / 2 4- A 3 f 3 = 0, 
G 4- A^r 4- A 1 g 1 4- A 2 y 2 4- A 3 g 3 = 0, 
H 4- A/i 4" Aj/ij 4~ A 2 A 2 4" A 3 h 3 —— 0. 
Hence representing as before the moments of the pairs of lines by 01, 02, &c., 
we deduce 
. AjOl 4" A 2 02 4" A 2 Q3 = 0, 
AIO4- . 4- A 2 12 4- A 3 13 = 0, 
A20 4- Aj21 . 4- A 3 23 = 0, 
A30 4- A^l 4- A 2 32 . = 0, 
so that, as already mentioned, we have 
01, 
02, 
03 
10, 
12, 
13 
20, 
21, 
23 
30, 
31, 
32, 
. 
as the condition that the four given lines may have a twofold tractor. 
Article Nos. 54 to 56. Hyperboloid passing through three given lines. 
54. The direct investigation is somewhat tedious; but I write down, and will 
afterwards verify, the equation of the hyperboloid passing through the three given lines 
(ftj, bi, Ci, f\, g u hi), (ft 2 , b 2 , c 2 , f 2 , g 2) h. 2 ), (ft 3 , b 3 , c 3 , f 3 , g 3 , h 3 ). 
Writing for shortness (agh), &c. to denote the determinants 
<h, 9u 
ft 2 , 9 2 > 
a», 9s, 
&c. 
K 
h 3