NATURE 
193 
THURSDAY, DECEMBER 28, 1882 
MATHEMATICS IN AMERICA 
American Fournal of Mathematics, Pure and Applied. 
Published under the Auspices of the Johns Hopkins 
University. Vols. III. and IV. (Baltimore: Isaac 
Friedenwald.) And other Mathematical Journals. 
HE American Fournal of Mathematics was estab- 
lished in 1878 under the auspices of the Johns Hop- 
kins University at Baltimore, and four handsome quarto 
volumes of 400 pages each have now been published. 
Prof. Sylvester was editor-in-chief of the first three 
volumes, being assisted by Mr. Story as editor-in-charge, 
but the last volume bears Sylvester's name alone as 
editor. 
A notice of the first two volumes of the Journal ap- 
peared in NATURE, vol. xxii. p. 73, and the hope was 
there expressed that it might have as great a future be- 
fore it as awaited Crelle’s Journal half a century before. 
A careful examination of the last two volumes shows that 
the promise of the earlier volumes has been so far main- 
tained, and that the Journal has already acquired a distinc- 
tive character of its own. It almost invariably happens that 
mathematical journals exhibit marked characteristics, and 
that certain branches of the subject occupy a pre-eminent 
position. One paper leads to another relating to the 
same questions, and the original bias of a journal is gene- 
rally due, both directly and indirectly, to its editor, as 
authors naturally prefer to send contributions where they 
are more likely to be understood and appreciated. That 
this is especially the case with the American Journal is 
what we should expect, as besides being the principal 
contributor, the editor is professor in the institution with 
which it is connected, and many of the papers are by his 
former pupils and colleagues. Although a very distinct 
tendency is thus evident in the direction of the large 
group of subjects (and more particularly Higher Algebra 
and Higher Arithmetic) with which the name of Prof, 
Sylvester is associated, it is not to be supposed that the 
Journal has become narrow in its scope. On the con- 
trary, the whole range of mathematical subjects is very 
fairly represented, as will appear from the following para. 
graphs, which contain a list of the papers in vols. iii, and 
iv., an attempt being made to group them to some extent 
according to subjects. 
The arithmological papers are numerous. Prof. Sylvester 
gives closer limits for a quantity which occurs in 
Tchebycheff’s well-known investigation of the number of 
primes inferior to any given prime ; he contributes also a 
note on the trisection and quartisection of the roots of unity, 
and an instantaneous proof of a theorem of Lagrange’s on 
the divisors of a certain quadratic form. In a paper 
on a point in the theory of vulgar fractions he gives a 
method of developing any vulgar fraction as a sum of cer- 
tain special fractions, each having unity as its numerator, 
This development he terms a sorites, and he remarks 
that it was suggested to him by the chapter in Cantor’s 
Geschichte der Mathematik, which gives an account of the 
singular method in use among the ancient Egyptians for 
working with fractions: it was their curious custom to 
resolve every fraction into a sum of simple fractions ac- 
VOL. Xxvil.—No. 687 
cording to a certain traditional method, which however 
only leads in a few simple cases to a sorites. There are 
two papers by Mr. O. H. Mitchell, both relating to the 
theory of congruences : one of them contains a generalisa- 
tion of Fermat’s and Wilson’s theorems. There is also a 
short note by Prof. Newcomb on the relative frequency of 
the occurrence of the digits as leading figures in logarith- 
mic tables. 
The contributions to the higher algebra occupy a very 
conspicuous place. The important tables of the gene- 
rating functions and ground forms of binary quantics 
which have been calculated by Prof. Sylvester and Mr, 
F. Franklin, with the aid of a grant from the British 
Association, are continued. Mr. Franklin is also the 
author of a separate paper, in which he gives a consecu- 
tive account of the methods, due to Cayley and Sylvester, 
of calculating the generating functions for binary quantics 
and thence determining the number of fundamental in- 
variants and covariants of any order and degree. Prof. 
Sylvester gives a determination of the impossibility of 
the binary octavic possessing any ground form of de- 
gree 10 and order 4. There is a paper on the 34 concomi- 
tants of the ternary cubic by Prof. Cayley, who also gives 
a specimen of a literal table for binary quantics and cer- 
tain tables for the binary sextic ; and there are some notes 
on Modern Algebra by M. Faa de Bruno, of Turin. Mr, 
Mitchell and Mr. T. Muir, of Glasgow, give theorems 
relating to determinants. 
Prof. Wm. Woolsey Johnson is the author of a paper 
on strophoids. The term strophoid has been applied by 
French writers toa cubic curve, the symmetrical form of 
which Dr. Booth discussed under the name of the Logo- 
cyclic curve. The author gives to the term a more extended 
signification, and defines a strophoid as the locus of the 
intersection of two straight lines which rotate uniformly 
about two fixed points ina plane. Dr. Booth’s curve is 
included as a particular case of the class of curves which 
Prof. Johnson terms right strophoids. Prof. Sylvester 
considers the theory of rational derivation on a cubic, and 
Mr. Story is thus led to discuss the subject more fully in 
a separate memoir: the points on the curve which are 
considered are those whose coordinates can be expressed 
as rational functioms of an arbitrary initial point on the 
curve. Mr. Samuel Roberts contributes a paper on the 
generalisation of local theorems, in which the generating 
point divides a variable linear segment in a constant ratio ; 
and there is a note by Miss Christine Ladd on segments 
made on lines by curves. 
There are three papers on solid geometry, all by Mr. T. 
Craig: they relate to the orthomorphic projection of an 
ellipsoid upon a sphere, to certain metrical properties of 
surfaces (in 7 dimensions as well as in three dimensions), 
and to the counter-pedal surface of an ellipsoid. The 
surface which the author designates the counter-pedal is 
the locus of the intersections of central planes parallel to 
the tangent planes of the ellipsoid with the normals at 
the corresponding points of contact; its equation is worked 
out, and is found to be of the tenth order. Mr. E. W. 
Hyde contributes {a note on the centre of gravity of 
a solid of revolution, and there is a discussion by 
Prof. Stringham on the regular figures in #-dimensional 
space. 
Prof, Cayley gives a note on the analytical forms or 
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