104 
NATURE 
[Fune 4, 1885 
A, &, C, D, the four distances on a variable plane from the 
fixed points @,4,c,¢. I must promise that using quadri- 
planar co-ordinates x, y, z, ¢ analogous to those employed 
Just now for the plane—viz. such as will cause 
xty+e2e+f=0 (1) 
to become the equation to “the plane at infinity,” the 
sphere circumscribing the fundamental pyramid aéca 
takes the analogous form to that given for the circle from 
which indeed it may be deduced with a stroke of the pen 
—viz. the equation to this sphere will be— 
(abfxry + (acPxre+(adyxrt+ (dcfyz+(bd)yt 
+ (cd)*zt=o (2). 
Moreover the distances of a plane whose equation}is— 
Ax+By+Cz+Dt=o0 (3) 
from the vertices of the pyramid will be A, B, C, D each 
multiplied by the same known quantity. 
The intersection of the plane at infinity with any sphere, 
and consequently with the circumscribing sphere named, is 
“the circle at infinity ;” hence if / is the constant function 
required we may find it as the function of 4, B,C, D, 
which becomes zero when the plane (3) is tangential to 
the intersection of the plane (1) with the sphere (2), or, 
which is the same thing, when the intersection of the 
planes (1) and (3) is tangential to the sphere (2), and this 
function is well known to algebraists to be the determinant 
formed by bordering the determinant to (2) with the co- 
efficients of (1) and (3), ze. we may take as the constant 
function / the determinant following :— 
e A B Cc D e 
A e (a6)? (ac? (ad) I 
B (6a)? @ (bc? (6d 1 
GY (Ga). (eb)? e (ed)? { 
D (da (db (do? I 
e I I I I e 
of which the developed value is easily found to be— 
— 3(a6)'!(C — D) + 23 (a 6)" (ac)*(B — D)(C — D) 
wi-pes (A-—C)(B-D 
+ 23 (2b)? (cd) ? (a. DB 26) 
This value of the constant function in its expanded form 
I some time ago found by a different method, and sent'in 
the shape of a question to the Educational Times.* In 
a brief correspondence which ensued with Prof. Cayley, he 
wrote to me giving the equivalent determinant form which 
he arrived at by a totally different order of conceptions 
and in a very beautiful manner, as follows. We may 
regard the differences between 4, 4, C, D as equal to the 
differences between the distances of a, 4,c,@ from a fifth 
point, e, at an infinite distance, and may call ae, de, ce, de 
equal to 4 + KX, B+ KH, C+AK4, V+ K respectively, 
where & is infinite. Hence by his own well-known 
theorem regarding mutual distances of five points we shall 
have— 
/ @ (ab)? (ac)? (ad)? (A+K) 1 
| (6a) ry (bc)? (6d)? (B+K)2 «| 
(¢ a)* (G 6)? @ (¢ d) (C+ Ky 1 | | 
(da)? (db)? (de)? e) Oats - 
(A+K)? (B4K)? (C+K): (D+K)? e I 
I Mt I I I @ 
And by the ordinary well-known rules in determinants 
* If the differences between 4, B, C, D be regarded as the minor deter- 
. aye eG £ 2) + 5 
minants of the bilinear matrix ~* = e , any practised algebraist would 
at once recognise that my form becomes expressible as a determinant of the 
6th order, and I think I could hardly have failed eventually to have made 
this observation, the more especially as I was aware of the connection of the 
subject with that of the section of any sphere with the plane at infinity—but 
as a matter of fact Cayley anticipated me, and was the first to actually write 
down the function under the form of a determinant. 
In each method the concept of infinity appears, but in mine that of the 
imaginary as well; and although more far-fetched than the other, the latter 
possesses the advantage of yielding the result as the transcript of a mere 
mental process without involving the necessity for the performance of any 
work whatever of algebraical reduction. 
for combining lines with lines and columns with columns 
it may easily be shown that the above determinant is of 
the form 47,K*+ G&-+ H, where F, represents— 
= 15 A B Cc 18) e 
A fa) (a6)? (ac? (ad 1 
B (6a) @ (bc (6d)? 1 
Cc (ca? (¢6) e@ (cd) I 
D (da) (db)? (dc? e@ I 
e I I I I e 
Consequently /, =o. This equation gives not only the 
form of the constant function but the value of the con- 
stant (4, when the element — 4 is suppressed, being iden- 
tical with my /). 
On removing the line and column of capital letters the 
above determinant equated to zero expresses the condition 
of the points a,4,c,@ lying in a plane—as proved by 
Cayley in days long past (and still ordinarily so proved) 
by a very artful manner of multiplying a determinant into 
a numerical multiple of itself; but this result follows as 
an instantaneous consequence of the reflexion that if 
a,b, c,d did not lie in a plane the above equation would 
mean that the circumscribing sphere was ¢ouched by the 
plane at infinity, whereas we know that this plane never 
touches but has the faculty of always cutting every sphere 
in a constant circle of imaginary points. Hence the ex- 
istence of this equation implies the coplanarity of the four 
points a, 4, c, d, and the converse proposition may be shown 
by simple algebraical reasoning to follow from this.* 
Postscrvtpt.—\ have been led by what precedes to a 
rather interesting observation in universal geometry. 
Suppose we form a determinant with the squared 
distances of one group of # points from another equi- 
numerous group any or all of which may be coincident 
with those of the former one: and to each line and at the 
foot of each column of this determinant affix a unit; a 
determinant so formed we may agree to call the bordered 
determinant of either group in regard to the other. 
Thus ex. 7. 
aad ae’ ay I 
ba* 6B by I 
ca cB Give I 
I I I 
is the bordered determinant of a,4,c in regard to a,By. 
When the two groups .are one group repeated we may 
call this determinant the bordered se/-determinant of the 
groups. 
My theorem is that the bordered determinant of two 
equi-numerous groups in respect to one another is a mean 
proportional to the bordered self-determinant of one of 
the groups, and that of the projection upon its zzveauw of 
the other group. [Thezveaw to’a group of points means 
the homaloid (Clifford’s flat) of the lowest number of 
dimensions which contains the group. ] 
We may regard a group of # points as the vertices of a 
figure whose squared content we know by Cayley’s 
theorem above referred to is a sub-multiple of the bordered 
self-determinant of the group; it is in fact that quantity 
divided by (—)**1 2" (1.2.3... #)*,so that we may vary 
the statement of the theorem and say that the product of 
the contents of the figures denoted by two equi-numerous 
groups into the cosine of the inclination of their mzveaus 
is a known numerical sub-multiple of the bordered deter- 
minant of one group in respect to the other. Thus, keep- 
ing at first within the limits of conceivable space, we see 
* The equation in the text extended to the points A, B,... Z, X assumes 
new importance and rises to philosophic interest when regarded as the izérvinsic 
equation to the ntveau of A, B,... L, in which the co-ordinates of the 
variable point Vin the niveau are the squares of A XY, BX,... 2X; itis 
of course an equation of the second degree in these co-ordinates. The dis- 
tances of either of two points from the other are the same in quantity but 
differ in sign. Hence the sgnave of either is the natural measure of the 
interval de¢ween the two points. 
