900] 
JAMES JOSEPH SYLVESTER. 
47 
There is, in 1844, in the Philosophical Magazine, a valuable paper, “ Elementary 
Researches in the Analysis of Combinatorial Aggregation,” and the titles of two other 
papers, 1865 and 1866, may be mentioned: “Astronomical Prolusions; commencing* 
with the instantaneous proof of Lambert’s and Euler’s theorems, and modulating 
through the construction of the orbit of a heavenly body from two heliocentric distances, 
the subtended chord, and the periodic time, and the focal theory of Cartesian ovals, 
into a discussion of motion in a circle and its relation to planetary motion ”; and 
the sequel thereto, “Note on the periodic changes of orbit under certain circumstances 
of a particle acted upon by a central force, and on vectorial coordinates, &c., together 
with a new theory of the analogues of the Cartesian ovals in space.” 
Many of the later papers are published in the American Mathematical Journal, 
founded, in 1878, under the auspices of the Johns Hopkins University, and for the 
first six volumes of which Sylvester was editor-in-chief. We have, in vol. I., a 
somewhat speculative paper entitled “An application of the new atomic theory to 
the graphical representation of the invariants and covariants of binary quantics,” 
followed by appendices and notes relating to various special points of the theory; 
and in the same and subsequent volumes various memoirs on binary and ternary 
quantics, including papers (by himself, with the aid of Franklin) containing tables of 
the numerical generating functions for binary quantics of the first ten orders, and for 
simultaneous binary quantics of the first four orders, &c. The memoir (vols. II. and 
III.) on “Ternary cubic-form equations” is connected with some early papers relating 
to the theory of numbers. We have in it the theory of residuation on a cubic 
curve, and the beautiful chain-rule of rational derivation; viz. from an arbitrary 
point 1 on the curve it is possible to derive the singly infinite series of j)oints 
(1, 2, 4, 5, ...,3p±l) such that the chord through any two points, m, n, again meets 
the curve in a point m + n, m ~ n (whichever number is not divisible by 3) of the 
series; moreover, the coordinates of any point m are rational and integral functions 
of the degree m 2 of those of the point 1. 
There is in vol. v. the memoir, “ A Constructive Theory of Partitions arranged 
in three acts, an Interact in two parts, and an Exodion,” and in vol. VI. we have 
“ Lectures on the Principles of Universal Algebra,” (referring to a course of lectures 
on multinomial quantity, in the year 1881). The memoir is incomplete, but the 
general theories of nullity and vacuity, and of the corpus formed by two independent 
matrices of the same order, are sketched out; and there are, in the Comptes rendus 
of the French Academy, later papers containing developments of various points of the 
theory,—the conception of “ nivellators ” may be referred to. 
The last-mentioned paper in the American Mathematical Journal was published 
subsequently to Sylvester’s return to England on his appointment as Savilian Professor 
of Mathematics at Oxford. In December 1886, he gave there a public lecture 
containing an outline of his new theory of reciprocants (reported in Nature, January 7, 
1887), and the lectures since delivered are published under the title, “Lectures on 
the Theory of Reciprocants” (reported by J. Hammond), same Journal, vols. vm. to 
x.; thirty-three lectures actually delivered, entire or in abstract, in the course of