578 
NATURE 
[Oct. 15, 1885 
provided that 7, 7,7; \, yw, v are correlated systems of 
coordinates. 
Images: Reciprocats or Polar Reciprocals 
lt is very convenient to speak of any function which 
equated to zero expresses a figure as an zage* of such 
figure ; thus ex. gv. Ex + ny + G may be spoken of as an 
image of the line &, n, ¢ and of the point 2, y, z. 
A curve being the concept common to a Jocus and an 
assembly (the common ground, so to say, of the existence 
of each of them), will be capable of being imaged in 
terms of either direct or inverse coordinates. If the two 
coordinate systems are supposed to be correlated (as they 
ought always to be) then any two homogeneous functions 
which are reciprocal, or, let us say, conjugate to one 
another (each in common parlance the polar reciprocal of 
the other) will be images—the one of the curve under its 
aspect as a /ocus, the other of the very same curve under 
its aspect as an assemblage. 
Reduced Perpendicular Distances 
An extremely convenient system of homogeneous co- 
ordinates of a point is where each coordinate is the 
distance from one of the boundaries of the fundamental 
enclosure divided by the distance of that boundary from 
the opposite angle. Such coordinates may be termed 
coordinates of reduced distance or reduced coordinates ; 
they are analytically defined by their sum being unity. 
If a, be the two vertices which correspond to the co- 
ordinates of reduced distances, the squared distance of 
any two points, x y, z,...; x, y’ 2',... in extension of 
any order is capable of being expressed by the formula 
3 (ab)? (x — x') (y' — 9), which, as far as I have been able 
to ascertain, is nowhere stated in the books, except for 
the case of trilinear coordinates. 
Exchangeable Figures 
Two figures indistinguishable from each other by any 
of their internal properties, but incapable of occupying the 
same place (such as the left- and right-hand glove or shoe) 
have received the very awkward and misleading name of 
symmetrical figures ; I propose to call them exchangeable 
figures, inasmuch as in the nature of things, as they are 
in themselves (without regard to the limitation of the 
human faculties), they may be made to pass into eachother’s 
places by a semi-revolution about a suitable homaloidal | 
axis. 
The Point-Pair at Infinity, Lines and Planes of Null 
It has been already shown in these columns that the 
“absolute” in a plane has full right to be called the | 
point-pair at infinity, in analogy with the received ex- 
pression of the /zve at iufinity, and those who have con- 
sidered what has been here stated under the head of 
reciprocity will see good grounds for admitting that the line | 
at infinity ought to be regarded as a complete line, 7.2. as 
made up of two analytical “ semi-droites.” 
Every line through either half of the absolute besides 
the property of being infinitely distant from any point in 
the finite region may be termed a “ne of null, in the 
sense that the distance between any two points in such 
line is zero. 
In like manner any plane /fouching the absolute in ex- 
tension of the 3rd order, besides being infinitely distant 
from the finite region, is in the same sense a plane of 
null; in it, form is divorced from content, for a figure of 
any shape being described upon such plane, its content 
will be 227. 
Plurt-duality : Contatning and Contained 
In extension of z dimensions each continuum of A 
dimensions stands in a relation of reciprocity to one of 
* When an zmage is given, its object is absolutely determined, but not 
vice versa, Since an image may ke magnified or diminished at will by the 
introduction of a constant factor. 
Z—\—1 dimensions, the total number of these- “ dual- 
z+1 zi 
CAC . s z am 
ities ” being when z is odd and > when 7 is even 
I 
being its own reciprocal). It is very convenient in 
connecting reciprocal geometrical statements to ignore 
the difference between (and to regard as exchange- 
able and equivalent) the terms contazning and contained 
7 as applied to heterogeneous continua; indeed the 
ordinary distinctive use of these words suggests an 
erroneous conception ; as ex. gv. of a line being made 
up of points or a plane of lines. A point may be said to 
contain every line or plane which passes through it, and 
a line every point which lies on it, and every plane which 
passes through it: as an example of this extended locu- 
tion the order, rank, and class of a surface may be defined 
as follows—viz. the order and class as the number of its 
point and plane elements respectively contained in any 
given line; the rank as the number of its line elements 
contained in common by any given point and plane which 
contain one another. 
A plane-section of a surface is the totality of its point- 
or line-elements contained in a plane and similarly a 
point-section (an enveloping cone), the totality of its 
plane- or line-elements contained in a point : hence in- 
differently the class of any plane-section or the order of 
any point-section of a surface is its rank.* 
J. J. SYLVESTER 
NOTES 
ALL the five French academies will celebrate by a banquet 
the ninetieth anniversary of the foundation of the Institut, which 
was established on October 25, 1795, by the Conseil Leégislatif 
and Directoire Executif of the French Republic. The actual 
organisation is not quite the same as the original, great altera- 
tions having been made in 1814, and only partially abolished on 
subsequent occasions. 
THE death took place last month of General J, J. Baeyer, 
President of the Central Bureau for European Triangulation and 
of the Royal Prussian Geodetic Institute. General Baeyer had 
reached the age of ninety-one years. A biography of some length 
will be found in the Astronomische Nachrichten, No. 2687. 
M. RoBIN, a member of the Paris Academy of Sciences and of 
the French Senate, died last week. He had devoted his exertions 
to microscopy, and was professor to the School of Medicine. 
* The word s/read, to signify an unlimited expanse of discontinuous 
points and so used by Dr. Henrici, is, Iam informed, originally due to the 
late Prof. Clifford. In ignurance of this fact, on hearing that Henrici had 
been attacked for his use of the word, I stated my belief that it must have 
been borrowed from my use of it to signify a limited portion of a tissue of 
equi-spaced points, such as that which is turned to so profitable account ingay 
constructive theory of partitions in the American Fournal of Mathematics. 
I did not know at the time that Clifford had used the word, nor that Dr. 
Henrici’s treatise preceded by several years the publication of my memoir 
above referred to. This erroneous oral statement seems to have found its 
way by some more or less circuitous channel to the columns of the Saturday 
Review ina notice of a criticism, by Mr. Dodgson, of Dr. Henrici’s geo- 
metrical manual in the Scientific Series. Dr. Ferrers (the Master of Caius 
College, Cambridge) was the first to apply a spread to demonstrate in- 
tuitively a celebrated arithmetical theorem of reciprocity due to Euler, 
Mr. Durfee a quarter of a century later led the way to a further and more 
pregnant use of the same by showing how to trisect a symmetrical spread 
bounded by two right lines and a broken line into a regular square and 
two quasi-triangular appendages, to which I superadded the notion of 
mult.secting it into a succession of angles. Another pupil of mine at the 
Johns Hopkins University (Mr. Ely) has laid the foundation of a new 
theory of partitions, by studying the various modes of decomposing a solid 
spread of discontinuous points ; his memoir on the subject is to be found ina 
recent volume of the American Mathematical ¥ournal. 
By means of the trisection method I obtained zxter alfa a new expansion 
of (1 — v2)(1 — 27 . (1 — x72), which, on making =< unity and # infinite 
leads immediately ‘to Buler’s celebrated pentagonal-power series, and other 
results of a totally novel kind by the multisection method : so that a spread 
may justly be regarded as a potent instrument or magical mirror for extending 
old and bringing to view new truths in the wonderland :of partition and 
elliptic-function series. 
