96 
ON THE SIX COORDINATES OF A LINE. 
[435 
77. As to the interpretation of these formulae, taking 
ABCD as the fundamental tetrahedron for (x , y , z , w), 
AqBqCqDq „ „ (x 0 , y<>> z 0> Wo), 
then 
(A x , y x , v x , pifix 0 , y 0 , z 0 , 
W 0 ) = 0 
is the equation of plane 
BCD, 
(\ 2 , y 2 , v 2 , p 2 ][ „ 
)— 0 
)9 
CD A, 
(A 3 , /¿3, v 3 , p 3 7£ „ 
)=o 
97 
DAB, 
(A4, yi, Vi, ,, 
) = ° 
99 
ABC, 
whence, observing that the second and third equations belong to two planes each 
passing through the line DA, it appears that the coefficients 
yv v\ \y Ap yp vp 
23 ’ 23 ’ 23 ’ 23 ’ 23 ’ 23 ’ 
are the six coordinates of the line DA, expressed in regard to the tetrahedron 
(A 0 B 0 CoD 0 ) ; and similarly that the coefficients in the six expressions of the trans 
formation formula are the six coordinates of the lines AD, BD, CD, BC, CA, AB 
respectively in regard to the tetrahedron (A 0 B 0 C 0 D 0 ). 
In the preceding formulae for the transformation of coordinates the ratios only have 
been attended to, no determinate absolute magnitudes have been assigned to the 
coordinates (a, b, c, f, g, h). But I will nevertheless show how we may attribute 
absolute magnitudes to these coordinates. 
Article Nos. 78 to 80. New definition of the six coordinates as to their absolute 
magnitudes. 
78. I assume (x, y, z, w) to be “volume” coordinates; viz. taking as before ABCD 
for the fundamental tetrahedron, and denoting the point (x, y, z, w) by P, I assume 
that we have 
x : y : z : w : 1 = PBCD : ABCD : ABPD : ABCP : ABCD, 
where PBCD, &c. denote the volumes of the several tetrahedra PBCD, &c. It is to 
be noticed that the volume is in every case taken with a determinate sign : analytically 
points A, &c. and writing 
PBCD = 
Va, 
z a)> 
&c., 
as 
Xp, 
X b , 
OCc, 
Xd 
Vp> 
Vo, 
Vd 
> 
z b, 
Zc, 
Zd 
1 , 
1 , 
1, 
1 
&c. 
(whence of course PBCD — PCD A = — PCBD, &c. according to the rule of signs): or 
we may in an equivalent manner, but less easily, determine the sign, by considering 
the sense of the rotation about CD (considered as an axis drawn from C to D) which 
would be produced by a force along PB (from P to B).