XXV11 



that domain of mathematical science which is in any way connected 

 with algebraical form. Among the first to enter the field was 

 Sylvester, then living in London ; he and Cayley were in constant 

 communication, alike oral and written, and carried on their work in 

 the most friendly relations with one another. Boole also resumed his 

 investigations, and both he and Salmon made substantial additions to 

 the theory. The continental mathematicians also had begun their 

 important contributions, chief among them being Aronhold, Hesse, 

 and, at a later date, Hermite. Aronhold, indeed, devised the so- 

 called symbolical method, now the favourite method with German 

 workers ; in its origin it is nearly the same as the symbolical method 

 introduced by Cayley, but the subsequent developments due largely 

 also to Clebsch and to Gordan run on lines entirely different from 

 Cay ley's. 



After a time, Cayley began his series of ten memoirs on quantics ; 

 they must rank among the most wonderful combinations of original 

 researches and papers upon a single theory ever produced. They 

 contain a splendid exposition of the theory as already established ; 

 they are full of original contributions to the subject, and as they take 

 account of the work done by other authors, they have the further 

 interest of showing how the subject grew between the appearance of 

 the ' Introductory Memoir ' in 1854 and the appearance of the ' Tenth 

 Memoir ' in 1878. This is hardly the opportunity to write a history 

 of the subject by apportioning among the various investigators the 

 sections which they respectively originated ;* yet reference should be 

 made to two matters. 



First, one of the problems that greatly interested Cayley was the 

 determination of the complete asyzygetic system of irreducible in- 

 variants and covariants appertaining to a binary form, that is, the 

 system such that every invariant and every covariant of the form can 

 be expressed as a rational integral algebraical function of the 

 members of the system, the coefficients in the fnnction being- 

 numerical only. In his ' Second Memoir on Quantics '| he had 

 accurately determined the number (and their degrees) of the asyzy- 

 getic invariants for binary forms of orders 2, 3, 4, 5, 6 ; he had also 

 accurately inferred the number (together with their degrees and 

 their orders) of the asyzygetic covariants for binary forms of orders 

 2, 3, 4, all these concomitants being subsequently tabulated. But, in 



* Some information will be found in an appendix to Salmon's ' Lessons on 

 Higher Algebra; ' also in the notes and references at the end of the second yolume 

 (pp. 598 601) of the ' Collected Mathematical Papers.' A valuable and exhaustive 

 report, containing a full history of the subject, was drawn up by Prof. Dr. Franz 

 Meyer, and published under the title " Bericht fiber den gegenwartigen Stand der 

 Invariantentheorie " (' Jahresber. d. Deutschen Mathem.-Vereinigung," I, 1892). 



t ' C. M. P.,' vol. 2, No. 250; ' Phil. Trans.' (1856), pp. 101126. 



