ON THE SIX COORDINATES OF A LINE. 
93 
435] 
69. In the case of four given lines the condition (as noticed by Möbius) is that 
the four lines shall be generating lines of the same hyperboloid. In fact every line 
which meets thiee of the four lines must also meet the fourth line 5 for otherwise 
the moment of the system about such line would not be = 0. Calling the lines 
0, 1, 2, 3 and writing as before 01, 02, &c. for the moments of the several pairs of 
lines, then taking the moments of the system about the four lines respectively, we 
obtain directly the before-mentioned system of equations 
Ml + X 2 02 + X 3 03 = 0, 
X10 . + X 2 12 + X 3 13 = 0, 
X20 + X 1 21 . + X 3 23 = 0, 
X30 + Xj 31 + Xj 41 . —0, 
leading as before to the relation 
VoT V23 + V02 Vs! + VÖ3 VI2 = 0, 
and to the values 
X : \ : Xa : X* = Vl2 V23 Vitt : V23 V3Ö V<32 : V30 VÖI Vl3 : VÖI Vl2 \ f 20 
for the proportional magnitudes of the forces. These last equations give 
XXj 01 = X 2 X 3 23, 
which, representing each force by a segment on the line along which the force acts, 
denotes that the tetrahedron of any two of the forces is equal to the tetrahedron of 
the other two forces; this is in fact equivalent to the theorem of M. Chasles, that 
if a system of forces be in any manner whatever reduced to two forces, the tetra 
hedron formed by these two forces has a constant volume. 
70. In the case of five given lines, the lines must have a pair of tractors. Any 
four of the lines have in fact two tractors; and each of these tractors must meet 
the fifth line, for otherwise the moment of the system about the tractor would not 
be = 0. In the case where the four lines have a twofold tractor, the foregoing con 
sideration shows only that the fifth line meets the twofold tractor, but it fails to 
show that the twofold tractor is a twofold tractor in regard to the fifth line. 
71. I stop to consider this particular case under the present statical point of 
view. Taking the twofold tractor for the axis of z; let the line 0 meet this line in 
the point (0, 0, c), the coordinates (a, b, c, /, g, h) of this line being consequently 
(c cos ft -c cos a, 0, cos a, cos ß, cos y) 
and the like for the other four lines 1, 2, 3, 4. Using the sign 2 to refer to the 
last-mentioned four lines the equations of equilibrium become 
Xc cos ß + SXjCj cos ft = 0, 
Xc cos ci -(- Sx^Cj cos 0, ^ 
X cos a + X\ cos = 0, 
X cosß + l\ cos ft = 0, 
X cos 7 + 2X, cos 7j = 0.