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regard to the invariants of forms of order higher than 6 and the co- 

 variants of forms of order higher than 4, he came to the erroneous 

 conclusion that the respective numbers are infinite. The error was 

 not corrected until Gordan in his memoir,* dated 8th June, 1868, 

 and entitled ' JBeweis dass jede Covariante und Invariante einer 

 binaren Form eine ganze Function mit numerischen Coefficienten 

 einer endlichen Anzahl solcher Formen ist,' showed that the com- 

 plete system for a binary quantic of any order contains only a limited 

 number of members. Cayley at once returned to the question, and 

 having found a source of error (it was the neglected interdependence 

 of certain syzygies, reducing the numbers of invariants and co- 

 variants ; the interdependence had not previously been suspected), he 

 dedicated his ' Ninth Memoir on Quantics 'f (dated 7th April, 1870), 

 to the correction of the error and a further development of the theory 

 in the light of Gordan's results. His promptness in recognising and 

 giving immediate prominence to the work of the younger author 

 possibly prevented some controversy among unwise partisans ; it was 

 characteristic of the man. 



And, secondly, though his series of memoirs was brought to an end 

 with the tenth, his interest in the subject did not cease, and he fre- 

 quently wrote upon parts of it under other titles. In particular, 

 Captain P. A. MacMahon's discovery of a relation of a new character 

 between seminvariants and symmetric functions (viz., that the lead- 

 ing coefficients of the co variants of a binary quantic are the same as 

 the non-unitary partition symmetric functions of the roots of an 

 equation connected with a modified quantic) proved of the keenest 

 satisfaction to him. From time to time he wrote in the ' American 

 Journal of Mathematics ' upon this subject and upon symmetric 

 functions generally in this connection, always sympathetic and appre- 

 ciative of the advances made by others, able to grasp and assimilate 

 their ideas, but using them as a master and not as a follower. It was 

 not alone, however, to symmetric functions, upon which he had 

 written long and important memoirs as early as 1857, but to many 

 other cognate subjects that he extended his researches upon in- 

 variants and covariants. The theory of equations of the fifth and 

 higher degrees, Sturm's functions, Tschirnhausen's transformation, 

 partition of numbers, Arbogast's method of derivation, skew deter- 

 minants^ to quote no others are titles and subjects of papers, in 

 all of which are investigations of great value. The reason that they 

 are less known (if such be the case) than his other work in the same 

 line of ideas is perhaps due to the fact that the direct theory of in- 



* ' Crelle,' vol. 69 (1869), pp. 323354. 



t ' C. M. P.,' vol. 7, No. 462 ; ' Phil. Trans.' (1871), pp. 1750. 

 J His discoveries in this subject alone have done much to simplify the analytical 

 investigations connected with Pfaff's problem and the allied theory. 



