Oct. 15, 1885] 
understood, the term @ sface is susceptible of a more 
precise meaning than is usually attributed to it: its 
intrinsic equation is given by Cayley’s theorem of squared 
distances. It is a homaloid or flat of the 3rd as a plane 
is such of the 2nd, a line of the rst, and a point of the 
zeroth order. 
The phrase space of the 4th order ought accordingly to 
be superseded if we would avoid using the same word in 
two different senses—z.e. in a wider and narrower sense. 
Extension of the 4th order is the proper expression to 
take its place, and so in general we ought to speak of 
extension of any given order z, and drop the phrase 
space of 72-dimensions. 
Figure, Plasm, Enclosure 
A figure may exist in extension of any order. When 
pervasively limited by homaloids, simple and closed, I 
had proposed to give toit the Zrovészonal name of plasm, 
but Dr. Ingleby has supplied me with the more appro- 
priate, or at least more simple, term, eclosure. 
On the number and nature of simple regular enclosures 
in extension of any order, consult a remarkable memoir 
by Prof. Stringham* of the University of California 
(formerly of the Johns Hopkins University), in the third 
volume of the American Fournal of Mathematics. 
Homaloid, Flat, Niveau, Absolute Measure of Distance 
Homaloid, the term long ago introduced by the writer 
of this note, az, suggested by the late lamented Clifford, are 
now well understood, and need no new explanation ; but 
it is well to bear in mind the zztrinsic eguation which 
serves to define them Zo wit 
A homaloid in extension of the nth order is definable by 
means of an equation of the second order (naturally 
expressible in the language of determinants), in which 
(z+1) points are the standards of reference, and the 
squared distances from these of any other point in the 
homaloid are the coordinates. 
Observe that the squared length is the absolute measure 
of distance de¢ween two points. The distances of each 
from the other are not equal but opposite quantities 
differing in algebraical sign. 
A niveau is a very convenient term to signify the 
homaloid of the dowest order that can be drawn through 
a given point-group and is always wazgue; the order 
of the homaloid which is the #zveaz toa group of 2 points 
cannot exceed w — I. 
Curves, surfaces, &c., of the 1st, 2nd, and mth znd. 
A plane (or simple) curve is of the first kind ; “a twisted 
curve,” “ courbe gauche,” or a curve in extension of the 
3rd order, of the second kind, and in general a curve in 
extension of the 7th order is a curve of the (#—1)th 
kind. 
Similarly we may define a simple surface as one of the 
first kind, and a surface in extension of the zth order as 
one of the (#—2)th kind; and so in general a figure of 
variety 7+ (¢ being 1 for a curve, 2 for a surface), in exten- 
sion of the order 7, is one of the (z—z)th kind.f 
* Mr. Stringham, a native of ‘‘the bloody land” of Kansas, studied 
mathematics and fine art under Peirce and Norton, at Harvard, obtained a 
fellowship at the Johns Hopkins University, and completed his studies under 
Klein in Leipsic. In his memoir he has given perspective drawings of the 
bounding solids about a vertex of the regular figures in quaternary extension, 
such solids being supposed to be previously rotated round the vertex into the 
same sface, which of course may be done just a3 the bounding planes about 
a vertex of a regular figure in ternary extension may be rotated round that 
point into the same A/ane. 
+ A curve may be called a one-dimensional, a surface a two-dimensional, a 
solid a three-dimensional cox¢inuzme and, andsoon. Thus a solid is to a 
space what a surface is to a plane and a curve to a right line. 
t The ordinary systems of geometry, whether Euclidian or Non-Euclidian 
(Ultra-Euclidian would be the more correct term), contemplate figures as 
contained in homaloids of some order or another ; but this limitation has an 
empirical origin, and is not an essential ingredient of the pure theory of 
form ; for instance, a curve, Ze. a unidimensional continuum, may, and in 
general will, be such as cannot be contained in a homaloid of any number of 
dimensions whatever ; it might be said that the order of its #/veax in such 
case is infinite; but this would be a mere verbal quibble—the right view 
NALRGRLE 
S47 
Curve, Locus, Assembly, Envelop, Environment 
A curve is that which is common to a locus of points 
and an assemblage of tangents ; the locus is the exvelop of 
the assembly, and the assemblage the exvzronment of the 
locus. 
Lines and Points 
A line may be used in the double sense of a locus or 
direction. In the latter signification an Euclidian or 
objective line is the union of two lines running in contrary 
directions and an analytical line is a half-line, a “ semi- 
droite,” meaning, of course, a half-Euclidian line. 
So a point may mean either a position or an infinite 
assembly of lines (containing or) contained in it ; used in 
the latter sense, it might temporarily be termed a fencz/- 
point. 
There are half or split points, as there are half or split 
lines. Thus the infinite extremities of the asymptotes to 
a hyperbola are half-points, the union of two of them 
being the correspondent to a single point in any ellipse 
of which the hyperbola is a perspective image. 
Coordinates, Homogeneous and Correlated 
Homogeneous systems of coordinates may be distin- 
guished into absolute and proportional. 
In the former the absolute magnitudes of each are 
material, in the latter their ratios only. 
Also into direct and inverse. 
Direct coordinates are measured by given multiples of 
the distances of a variable point from fixed homaloids ; 
inverse by given multiples of the distances of a variable 
line, plane, &c., from fixed points. 
Correlated systems of direct and inverse coordinates 
are those in which my “universal mixed concomitant” 
(Clebsch’s connex) Ex-+ ny + (2 (for greater clearness I 
confine myself for the moment to a particular diagram- 
matic case) equalled to zero expresses a line whose inverse 
coordinates are &, 7, ¢, when these are made constant and 
a point (pencil-point) whose direct coordinates (when it 
is regarded as denoting position) are x, y, 2 when these 
in their turn are made constant. 
If the distances of a point from the sides of the triangle 
of reference are /, 7, 2, and of a line from the angles of 
the same triangle \, p,v, and if the direct coordinates 
being c/, dm, en, and the inverse ones yA, dp, ev, and the 
distances of the angles from the sides 7, g, »— 
cyp = aog = cer. 
1, 7m, 23; r, p, v are correlated systems. 
If Z' ni! n' p'; 1, m,n, p the direct coordinates of two 
corresponding points in a homography are connected by 
the Matrix J7 and )'p'v' wr’; A, pw, v, 7 (the inverse coor- 
dinates of two corresponding planes of the same homo- 
graphy) by the Matrix 47’, then if the two systems of 
coordinates are correlated, JZ and JZ’ will be opposite 
matrices.* 
Of course the like will be true in extension of all 
orders: thus ex. gv. in the case of a plane if for a given 
homography 
Zl’: alt+bm-+cn 
:m': dl+em+fn 
ia’: gl+hm-+kn 
Ns (ch — fh) + (fe — dh) p+ (dh — eg)v 
tip’: (ch — 6k) X+ (ak — cg) p+ (bg — ah)v 
zi: (Of — ce) X+ (ed — af) p+ (ae — bd)v 
being that it is sams niveau. The radical distinction therefore is not 
between the common Euclidian geometry and its generalisation (the so- 
called Non-Euclidian) but between the Homaloidal and the Anhomaloidal 
geometries. 
* In other words, for two point line, point-volume, &c., schemes homo- 
graphically related, employing correlated systems of proportional co- 
ordinates, the matrix which serves to express the relation between the direct 
coordinates of the first scheme and those of the second may be taken the 
transverse of the matrix which does the same between the inverse coordinates 
ot the second and those of the first. This is an important and as far as I am 
aware a new ¢heovert. 
