435] 
ON THE SIX COORDINATES OF A LINE. 
95 
75. There is another statical representation of the involution of six lines. If a 
system of forces act on a solid body, then taking six lines at random, the system 
will be in equilibrium if the sum of the moments be =0 in regard to each of the 
six lines. But if the six lines be in involution; then, for the very reason that a 
rotation about one of these lines is resolvable into rotations about the other five lines, 
if the sum of the moments be = 0 for each of the five lines, it will also be — 0 for 
the sixth line: that is, it is not sufficient for the equilibrium of the forces that the 
sum of the moments shall be = 0 for each of the six lines. And we thus see that 
six lines in involution are lines such that the equilibrium of a system of forces about 
each of the six lines as axes does not insure the equilibrium of the system. 
Article Nos. 76 and 77. Transformation of Coordinates. 
76. Reverting to the general definition of the six coordinates (a, b, c, f g, h) of 
a line by means of the points (a, /3, y, 8) and (a', /3', y', 8') on the line ; suppose that 
instead of the original coordinate planes x — 0, y — 0, z = 0, w = 0 (forming a tetrahedron 
ABCD) we have new coordinate planes ¿e 0 =0, y 0 = 0, z 0 = 0, w 0 — 0 (forming a tetrahedron 
A 0 B 0 C 0 D 0 ); and that the relations between the two sets of current coordinates are given 
by the equations 
x : y : z : w= (X 1} g u v lt pf^Xo, y 0 , z 0 , w 0 ) 
• P*2 ) V2> Pz^lAo) 2/o> Zq, W 0 ) 
• (Au /^3» v 3> P&Xo, yo> ^0» W 0 ) 
: (At, yU. 4 , Vi, pifjx 0, yo> Zo, W 0 ), 
with, of course, the like relations between the original coordinates (a, /3, y, 8) and new 
coordinates (a 0 , /3 0 , y 0 , 8 0 ), and between the original coordinates (a, /3, y, 8) and the 
new coordinates (a 0 ', ¡3 0 ', y/, 8f), of the two points on the line (a, b, c, f g, h); then 
taking (a 0) b 0 , c 0) f 0 , g 0> h 0 ) as the new values of the six coordinates of the line, viz. 
writing 
a 0 : b 0 : c 0 : f 0 ’• 9» . : K 
= Po^o — Po'yo • 7o«() /— 7o /(2 o : a ofio'~ a ofio '• »0^0 a 0 K '• fto 8f, —/3 0 §o ■ 7oSo — 7o ^0, 
we obtain a system of formulae which njay be conveniently written as follows. 
gv 
23' 
■ f ■ 9 
vX , 
, + 23 6 ° 
: h 
+ 23 1 
+ ^/o + 23^ + 23^ 
23" 
: 31 
: 12 
: 14 
: 24 
: 34 
viz. the top line stands for (g 2 v s — /¿s^) № o + — v 3 \) b 0 + &c., and the other lines aie 
obtained from this by mere alterations of the suffixes.