94 
ON THE SIX COORDINATES OF A LINE. 
[435 
These equations give 
SXjCj cos /9 T _ c cos /3 _ 
2A,,cosoei cos a 5 
we may without loss of generality take the homographic conditions which express that 
the axis of ^ is a twofold tractor of the four lines to be 
Cj cos /5?! _ c 2 cos /?, _ c 3 cos /3* _ c 4 cos j3 4 _ j 
cos a x cos ol 2 cos a 3 cos a 4 
and this being so, the last-mentioned equation becomes 
c cos 6 , 
= k; 
cos a 
and it thus appears that the axis of z is a twofold tractor in regard also to the line 0. 
72. In the case of six lines such that there exist along them forces which are in 
equilibrium, taking this as a definition of the involution of six lines, we may very 
readily obtain from statical considerations the before-mentioned construction of the sixth 
line; viz. it may be shown that given any five of the lines, say the lines 1, 2, 3, 4, •"> 
and a point 0, we can through the point 0 determine a plane il, such that any 
line whatever through the point 0 and in the plane il is in involution with the five 
given lines. Consider the tractors of any four of the lines, say the lines 2, 3, 4, 5 ; 
we may through the point 0 draw a line OA meeting the two tractors; that is, the 
lines 2, 3, 4, 5 and the line OA will have a pair of common tractors. There con 
sequently exist along these lines forces which are in equilibrium; and since only the 
ratios are material, the absolute magnitude of the force along the line OA may be 
anything whatever. Similarly, considering the tractors of the lines 1, 3, 4, 5, and through 
0 a line OB meeting these tractors, then there exist along the lines 1, 3, 4, 5 and 
the line OB forces which are in equilibrium, and the absolute magnitude of the force 
along the line OB may be anything whatever. Hence, combining the two sets of 
forces, we have, along a line through 0 in the plane OA, OB, but otherwise indeter 
minate in its direction, a force in equilibrium with forces along the lines 1, 2, 3, 4, 5 ; 
that is, the line found as above is a line in involution with the lines 1, 2, 3, 4, 5. 
73. It is to be added, that through 0 we cannot, out of the plane OA, OB, draw 
a line in involution with the lines 1, 2, 3, 4, 5 ; for if any such line OK existed, 
then we should have along each of the lines OA, OB, OK forces in equilibrium with 
forces along the lines 1, 2, 3, 4, 5; and the magnitudes of the three forces being 
each of them anything whatever, it would follow that along any line whatever through 
the point 0 there would exist a force in equilibrium with forces along the lines 
1, 2, 3, 4, 5; that is, any line whatever through the point 0 would be a line in 
involution with these lines. 
74. It hence appears, that drawing OA to meet the tractors of 2, 3, 4, 5; OB 
to meet those of 3, 4, 5, 1; OC to meet those of 4, 5, 1, 2; OB to meet those of 
5, 1, 2, 3; and OE to meet those of 1, 2, 3, 4; the lines OA, OB, 00, OB, OE will 
be in one plane, say the plane il: and that any line through 0 in the plane 14 will 
be a line in involution with the lines 1, 2, 3, 4, 5.