2l8 



NA TURE 



\yan. 3, 1889 



also considerable. They relate chiefly to finite analysis, 

 and cover by their subjects a large part of it : algebra, 

 determinants, elimination, the theory of equations, par- 

 titions, tactic, the theory of forms, matrices, recipro- 

 cants, the Hamiltonian numbers, &c. ; analytical and 

 pure geometry occupy a less prominent position ; and 

 mechanics, optics, and astronomy are not absent. A lead- 

 ing feature is the power which is shown of originating a 

 theory or of developing it from a small beginning ; there 

 is a breadth of treatment and determination to make 

 the most of a subject, an appreciation of its capabilities, 

 and real enjoyment of it. There is not unfrequently an 

 adornment or enthusiasm of language which one admires, 

 or is amused with : we have a motto from Milton, or 

 Shakespeare ; a memoir is a trilogy divided into three 

 parts, each of which has its action complete within itself, 

 but the same general cycle of ideas pervades all three, 

 and weaves them into a sort of complex unity ; the 

 apology for an unsymmetrical solution is — symmetry, like 

 the grace of an eastern robe, has not unfrequently to be 

 purchased at the expense of some sacrifice of freedom 

 and rapidity of action ; and, he remarks, may not music 

 be described as the mathematic of sense, mathematic as 

 the music of the reason .? the soul of each the same ! &c. 

 It is to be mentioned that there is always a generous 

 and cordial recognition of the merit of others, his fellow- 

 workers in the science. 



It would be in the case of any first-rate mathematician 

 — and certainly as much so in this as in any other case — 

 extremely interesting to go carefully through the whole of 

 a long list of memoirs, tracing out as well their connection 

 with each other, and the several leading ideas on which 

 they depend, as also their influence on the development 

 ef the theories to which they relate ; but for doing this 

 properly, or at all, space and time, and a great amount of 

 labour, are required. Short of doing so, one can only 

 notice particular theorems— and there are, in the casa of 

 Sylvester, many of these, " beautiful exceedingly," which, 

 for their own sakes, one is tempted to refer to— or one 

 can give titles, which, to those familiar with the memoirs 

 themselves, will recall the rich stores of investigation and 

 tlieory contained therein. 



A considerable number of papers, including some of 

 fihe earliest ones, relate to the question of the reality of 

 the roots of a numerical equation : in the several connec- 

 tions thereof with Sturm's theorem, Newton's rule for the 

 number of imaginary roots, and the theory of invariants. 

 Sylvester obtained for the Sturmian functions, divested of 

 square factors, or say for the reduced Sturmian functions, 

 singularly elegant expressions in terms of the roots, 

 viz. these were fj^x) = 2(rt — b)\x — c){x - d) . . . , 

 fix) = 2(« - b)\a - c)\b - c)\x ~ d) ...,8ic.; but not 

 only this : applying the Sturmian process of the greatest 

 common measure (not iof{x),f\x). but instead) to two 

 independent functions /{x), <t>{x), he obtained for the 

 several resulting functions expressions involving products 

 ef differences between the roots of the one and the 

 ether equation, /{x) = o, (f){x) = o ; the question then 

 arose, what is the meaning of these functions .'' The 

 answer is given by his theoiy of intercalations : they are 

 signaletic functions, indicating in v*rhat manner (when 

 the real ri)Ots of the two equations are arranged in order 

 «f magnitude) the roots of the one equation are inter- 



calated among those of the other. The investigations in 

 regard to Newton's rule (not previously demonstrated) are 

 very important and valuable : the principle of Sturm's 

 demonstration is applied to this wholly different question : 

 viz. X is made to vary continuously, and the consequent 

 gain or loss of changes of sign ii inquired into The 

 third question is that of the determinatidn of the charac- 

 ter of the roots of a quintic equation by means of in- 

 variants. In connection with it we have the noteworthy 

 idea oi facultative points ; viz , treating as the coordinates 

 of a point in ^-dimensional space those functions of the 

 coefficients which serve as criteria for the reality of the 

 roots, a point is facultative or non-facultative according 

 as there is, or is not, corresponding thereto any equation 

 with real coefficients : the determination of the charac- 

 ters of the roots depends (and, it would seem, depends 

 only) on the bounding surface or surfaces of the faculta- 

 tive regions, and on a surface depending on the dis- 

 criminant. Relating to these theories there are two 

 elaborate memoirs, " On the Syzygetic Relations &c.," 

 and " Algebraical Researches &c.," in the Philosophical 

 Transactions for the years 1853 and 1864 respectively; 

 but as regards Newton's rule later papers must also be 

 consulted. 



In the years j 85 1-54, we have various papers on homo- 

 geneous funjtions, the calculus of forms, &c. {Camb. am^ 

 Dub. Math. Journal, vols. vi. to ix.), and the separate 

 work "On Canonical Forms" (London, 1851). These 

 contain crowds of ideas, embodied in the new words, 

 cogredient, contragreaietit, concomitant., covariant, contra- 

 variant., invariant, e/naitant, coinbinant, cofnmutani, 

 canonical fortn, ple.xus, &c., ranging over and vastly 

 extending the then so-called theories of linear transform- 

 ations and hyperdeterminants. In particular, we have 

 the introduction into the theory of the very important 

 idea of continuous or infinitesimal variation : say that 

 a function, which (whatever are the values of the para- 

 meters on which it depends) is invariant for an infinitesimal 

 change of the parameters, is absolutely invariant. 



There is, in i84[, in the Philosophical Afaga2ine,?i 

 ) valuable paper, '• Elementary Researches in the Analysis 

 of Combinatorial Aggregation," and the titles of two 

 other papers, 1865 and 1866, may be mentioned ; " Astro 

 nomical 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 vec- 

 torial 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 

 Mathetnatical 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 en- 

 titled " An application of the new atomic theory to the 

 graphical representation of the invariants and covariants 

 of binary quantics," followed by appendices and nc|te3 

 relating to various special points of the theory ; and in 



