82 
ON THE SIX COORDINATES OF A LINE. 
[435 
it is clear that every line which meets each of the four given lines, will also meet the 
line (a, b, c, f, g, h) ; but the only lines meeting the four given lines are two deter 
minate lines, the tractors of the four given lines ; and the conclusion is, that the line 
(a, b, c, f g, h) is any line whatever which meets the two tractors. 
44. If, however, the four given lines have a twofold tractor, then the line (a, b, c,f g, h) 
is still a line having two conditions imposed upon it; it is in fact a line determined 
as in No. 21, viz. if on the tractor we take a series of points p, and through the 
tractor a series of planes P, corresponding homographically to the points, then the line 
(a, b, c, f g, h) is any line through a point p, in the corresponding plane P. 
45. Using as before 01, 02, ...12, &c. to denote the moments of the several pairs 
of lines, we have 
. A} 01 -f- A 2 02 -(- A 3 03 A 4 04 = 0, 
and thence also 
X10 . + A 2 12 + A 3 13 + A 4 14 = 0, 
A20 + Ai 21 . -f- A 3 23 4- X 4 24 = 0, 
A30 + A x 31 + A 2 32 . +A 4 34 = 0, 
A40 -f- Aj 41 + A 2 42 -|- X 3 43 . — 0, 
. 
01, 
02, 
03, 
04 
10, 
12, 
13, 
14 
20, 
21, 
23, 
24 
30, 
31, 
32, 
. 
34 
40, 
41, 
42, 
43, 
a relation between the moments satisfied in virtue of the original twofold relation ; but 
which, as a single equation, is of course not equivalent to the twofold relation. It is 
in fact easy to see that this equation expresses that the five lines have a common 
tractor; this is true, since in virtue of the twofold relation there are really two 
common tractors. 
I have not obtained from the linear equations any symmetrical expressions for the 
ratios A * Aj 1 X 2 1 A 3 . 
Case of a onefold relation. 
a , b , 
c, /, 
9 > 
h 
otj, bi, 
c i 1 /1, 
di. 
K 
, b%, 
C 2, /2, 
9a, 
K 
b 3 , 
C 3, fä> 
9», 
h 3 
«4, 6 4 , 
^4, fii 
9*, 
K 
a., b s , 
c s> /5, 
9a, 
K 
46. The onefold relation is