494 Mr, A. Cayley on a Surface of the Fourth Order. 
(if, as usual, i= “—1), or, what is the same thing, it is 
af + y* + (2+1)?=0; 
that is, the surface is made up of the two spheres, passing through 
the points A, B, C, and having seach of them the radius zero ; 
or say the two cone- spheres through the pots A, B,C. In 
other words, the equation 
BC,.AP+CA.BP+AB.CP= 
is the condition in order that the four points A, B, C, P may lie 
on a sphere radius zero, or cone-sphere. Using 1, 2, 8, din 
the place of A, B, C, P to denote the four points, the last-men- 
tioned equation becomes 
12,34+18.42414.23=0; 
and considering 12, &c. as quadratic radicals, the rational form 
of this equation 1s 
— 9 ee & 
Oo= 0, 19, 18, ie pein oi 
2] . ) 0 > 93, 24 
Shin Haier oQepiaide 
—ew) | ———en DB ——9 
47, 42, 48, 0 
In my paper “On a Theorem in the Geometry of Position,” 
Camb. Math. Journ. vol. uu. pp. 267-271 (1841), I obtained 
this equation, the four points being there considered as lying in 
a plane, as the relation between the distances of four points in a. 
circle, in addition to the relation 
a AO node 
Sn Ge aam o RROnramene 
iy ssT 
] 
which exists between the distances of any four points in a plane, 
The present investigation shows the signification of the equation 
O = 0 between the distances of four points in space; viz. it 
expresses that the four points he in a sphere radius zero, or 
cone-sphere. But the formula in question is im reality eluded 
in that given in the paper for the distances of five pomts in. 
space. Por calling the points 0, 1, 2, 3, 4, the relation between 
the distances of these five points is 
