Supplement, ~| 

 J unitary i8, 1894 J 



NATURE 



coveries, here set forth, which will be for ever linked to 

 the Professor's fame. 



The first volume comprises loo papers, numbered | 

 consecutively and nearly in chronological order, pro- 

 duced between the dates 1841 and 1853. The paper No. 

 13, " On the Theory of Linear Transformations " (1845), 

 marks a distinct epoch. Boole, in 1843, had proved the ' 

 invariantive property of the discriminant of a quantic 

 homogeneous in in variables, and Hesse, in 1844, had 

 established certain covariantive properties of the ternary 

 cubic. Here we find the general problem of " invariants " 

 proposed, and some progress made towards its solution. 

 The first step was the generalisation of Boole's theorem to 

 a quantic of order «, containing n sets of m variables, the 

 variables of each set entering linearly. This led to a 

 function of the coefficients which possesses Boole's in- 

 variantive property, but it was not the discriminant as 

 first pointed out by Ischlafli. This function, however, 

 was seen by Cayleyto necessarily satisfy a certain system 

 of partial differential equations, and he was thence led 

 to the capital discovery that a class of functions satisfied 

 the same equations, and that each member of the class 

 possessed the invariantive property. Other papers 

 followed, and finally, in the year 1854, Cayley commenced 

 the series of memoirs on Quantics which at intervals 

 during the succeeding five-and-twenty years appeared in 

 \.\\e Philosophical Traiisaclions oi \.\\& Royal Society. In 

 this way the theory of algebraic invariants was gradually 

 evolved. To this result there were many other con- 

 tributors, notably Sylvester (to whom most of the 

 nomenclature is due), Salmon, and Ham.mond in this 

 country, and Aronhold, Clebsch, Gordan, and Hilbert in 

 Germany. The first six memoirs are presented in vol. ii. 

 of the collection. In vol. i. may be noted also the theory 

 of conies of involution in connection with curves of the 

 third order ; the theory of " Pfaffiau? " ; and the theory 

 of surfaces of the third order. The last paper mentioned 

 — a very important one — was developed in a correspon- 

 dence with Dr. Salmon, to whom is attributed the 

 enumeration of the twenty-seven lines on the surface. In 

 vol. ii. we find in the notes which conclude the volume 

 an account of the early bibliography of the theory of 

 invariants ; also the remarkable memoir on the theory 

 of matrices, a subject which, at the present day, is ex- 

 hibiting considerable vitality. The memoir appears to 

 have been overlooked by mathematicians for more than 

 twenty )ears after it appeared in 1858. The single 

 exception appears to have been the paper by Laguerre, 

 '' Sur le Calcul des Systemes Line ures " (Jour. Ec. Polyt., 

 t. XXV.), in 1867. However, the subject was ultimately 

 taken up by Sylvester, in his " Lectures on Multiple 

 Algebra," in the American Journal of Mathematics, and 

 is now an important branch of pure mathematics, both 

 in its results and in its ideas. The volume is also 

 remarkable for researches on the " Partitions of 

 Numbers," " Skew Symmetric Determinants," the 

 '■'■ Theory of Groups," " Caustics," and " Curves of the 

 Third Order." 



Vol. iii. contains notably the contributions to dyna- 

 mics and astronomy. There is the valuable " Report on 

 the recent Progress of Theoretical Dynamics," from the 

 ^' Report of the British Association for the Advancement 

 of Science, 1857." The review extends from Lagrange, 



NO. 1264, VOL. 49] 



1788, to Bertrand, 1857. It, principally, gives an account 

 of the notable contributions of Lagrange, Poisson, Sir W. 

 R. Hamilton, and jacobi. There are papers on Lunar 

 Theory, Elliptic Motion, and the Problem of Three 

 Bodies ; also an extensive series of '' Tables of the 

 Development of Functions in the Theory of Elliptic 

 Motion." The paper (No. 221) "On the Secular Ac- 

 celeration of the Moon's Mean Motion " is interesting as 

 supplying an independent verification of Prof. Adams' 

 correction of Plana's expression for the true longitude. 

 Vol. iv. is chiefly remarkable for the " Report on the 

 Progress of the Solution of certain Special Problems of 

 Dynamics," from the Report of the British Association 

 for the Advancement of Science for the year 1862. At 

 the commencement the author adverts to a serious 

 omission in his former report in volume iii. This 

 has reference to the memoir by Ostrogradsky, " Mcmoire 

 sur les equations differentielles relatives au problcme des 

 Isoperimetres," Mem. de St. Pet. t. iv , 1850, which con- 

 tains, in the most general form, the transformation of 

 the equations of motion from the Lagrangian to the 

 Hamiltonian form, and also the transformation of 

 the system arising from any problem in the calculus of 

 variations to the Hamiltonian form. The remark is 

 also made that in a work by Cauchy, " Extrait du 

 Memoire presenle a I'Academie de Turin le 11 Oct., 

 1831," published in lithograph under date Turin 1832, 

 there is satisfactory evidence that Cauchy, in the year 

 1 83 1, was familiar with the Hamiltonian form of the 

 equations of motion. Mention is made of papers on 

 theoretical dynamics, notably by Bour, Jacobi, Natani, 

 Clebsch, and Boole, which had appeared subsequent to 

 the writing of the first report. 



The second report includes various problems relating 

 to the " particle " and the " solid body," " the problem 

 of three bodies," &c. A list of memoirs is added, and 

 increases the historical value of the report. One paper. 

 No. 265, "Addition to the Memoir on an Extension of 

 Arbogast's Method of Derivations," is published in this 

 volume for the first time. The general subject of 

 "quantics" is resumed in a seventh memoir, which is 

 principally devoted to ternary forms. An important 

 paper is that on the " Theory of Equations of the Fifth 

 Order." The quintic equation cannot be solved algebra- 

 ically. By the " resolution of the quintic " mathe- 

 maticians understand the expression of its roots in terms 

 of those of its resolvent sextic. During the fifteen years 

 preceding 1861 considerable progress had been made 

 with this problem by Cockle and Harley. In the year 

 mentioned Cayley obtained a new auxiliary sextic equa- 

 tion, and showed that the roots of the quintic are, each of 

 them, rational functions of its roots. A fresh impulse 

 was given to the subject by George Paxton Young and 

 Emory McClintock in vols. vi. and viii. of the American 

 Journal of Mathematics, 1884- 1886. The former indi- 

 cated the general form of the roots of the quintic, and 

 the latter made it possible to directly and visibly express 

 them. In a long note at the end of the volume Prof. 

 Cayley supplies the links between his own work and the 

 later work of McClintock, which he considers very im- 

 portant and remarkable. In vol. v. there are two papers 

 on the theory of curves in space, and in the" notes "the 

 Professor gives his present views, taking into considera- 



