313]
OX A SURFACE OF THE FOURTH ORDER.
69
In my paper “On a Theorem in the Geometry of Position,” Carnb. Math. Joarn. voh n.
pp. 267—271 (1841), [1], I obtained this equation, the four points being there con
sidered as lying in a plane, as the relation between the distances of four points in a
circle, in addition to the relation
1 ,
1 ,
1 ,
1
1,
0 ,
12 2
Ï3 2 ,
Ï4 2
1,
Si',
0 ,
23 2 ,
24
1,
3Ï 2 ,
32 2
0 ,
34 2
1,
il!
42,
I
05 ,1
0
which exists between the distances of any four points in a plane. The present investi
gation shows the signification of the equation □ = 0 between the distances of lour
points in space ; viz. it expresses that the four points lie in a sphere radius zero, or
cone-sphere. But the formula in question is in reality included in that given in the
paper for the distances of five points in space. For calling the points 0, 1, 2, 3, 4,
the relation between the distances of these five points is
0,
1 ,
1 ,
1 ,
1 ,
1
1,
0 ,
0?,
02,
Ö3 2
Ö4 2
1,
To\
0,
Ï2 2 ,
Ï3 2 ,
14'
1,
20,
21*
0 ,
23 2
24 2
1,
To\
31 2
32,
0 ,
34 2
1,
40*,
41,
42,
43 2 ,
0
Hence if 1, 2, 3, 4 are the centres of spheres radii a, ß, 7, 8, and if 0 is the centre
of a tangent sphere radius r, we have
01 = r + a, 02 = r ± ß, 03 = r ± 7, 04 = r + 8 ;
so that, for any given combination of signs, it would at first sight appear that r is
determined by a quartic equation; but by means of a simple transformation (indicated
to me by Prof. Sylvester) it may be shown that the equation for r is really a quadratic
one; moreover, the equation remains unaltered if the signs of a, ß, 7, 8 and of r, are
all reversed; and r 2 has thus in the whole sixteen values. In particular, if a, ß, 7, 8
are each equal 0, then r 2 is determined by a simple equation (r the radius of the
sphere through the four points); and if, moreover, r = 0, then we have for the relation
between the distances of the four points, the. foregoing equation □ = 0.
2, Stone Buildings, W.G., March 25, 1861.
