574 GEOMETRY. [790
And this diagram gives also the linear expressions of the coordinates (x 1 , y x , z x )
or (x, y, z) of either set in terms of those of the other set; we thus have
x 1 — ax + (3y + yz, x = ax 1 + a 'y x + a "z x ,
y, = a'x + ft y + y z, y = /3x t + /3 , y 1 + /3"z x ,
^ + P'y + l"z, z = yx 1 + y 'y 1 + y”z u
which are obtained by projection, as above explained. Each of these equations is, in
fact, nothing else than the before-mentioned equation p = a'% + /3'y + <y'%, adapted to the
problem in hand.
But we have to consider the relations between the nine coefficients. By what
precedes, or by the consideration that we must have identically x 2 + y 2 + z 2 = x^ + y 2 + zf,
it appears that these satisfy the relations—
o?
+
+ 7 2
= 1,
a 2 +
a' 2
+
a //2
= 1,
a! 2
+
/3' 2
4- y' 2
= 1,
P +
j8' 2
+
/3" 2
= 1,
a" 2
+
/3" 2
+ y" 2
= 1,
y 2 +
7 2
+
y" 2
= 1,
a a"
’ +
P/3"
+ 7' y"
= 0,
/3y +
Pi
' +
PY
= 0,
a"a
+
P'/3
+ 77
= 0,
ya +
iOL
+
// //
7 a
= 0,
a a
+
¡3/3'
+ 77
= 0,
a/3 +
a'p
' +
a "P'
= 0,
either set of six equations being implied in the other set.
It follows that the square of the determinant
a ,
/3 ,
7
a',
P,
7'
a,
(3",
//
7
is = 1; and hence that the determinant itself is = +1. The distinction of the two
cases is an important one: if the determinant is = +1, then the axes Ox 1} Oy 1} Oz x
are such that they can by a rotation about 0 be brought to coincide with Ox, Oy, Oz
respectively; if it is = — 1, then they cannot. But in the latter case, by measuring
Vi> ¿1 in the opposite directions we change the signs of all the coefficients and so
make the determinant to be = +1; hence this case need alone be considered, and it
is accordingly assumed that the determinant is = +1. This being so, it is found that
we have a further set of nine equations, a = /3'y" — /3"y, &c.; that is, the coefficients
arranged as in the diagram have the values
Py” - Pi
y'a" — y'a
a/3" — a''/3'
Py - (3y"
y"a — ya"
a"/3 - a/3"
f3y - Py
ya — y'a
a/3’ — a [3
