NA TURK 



189 



THURSDAY, JANUARY 1, 18S5 



THE "AMERICAN JOURNAL OF 



MA THEM A TICS 



American Journal of Mathematics, Pure and Applied. 



Published under the Auspices of the Johns Hopkins 



University. Vols, v., vi., vii.. Part I. (Baltimore : 



Isaac Friedenwald, 1SS2-4.) 

 T'HE general features of this Journal have been clearly 

 indicated in the notices of the previous volumes 

 (see X VTURE, vol. xxii. p. 73, vol. xxvii. p. 193), and we 

 need only remark under this head that these original 

 characteristics have been maintained throughout the 

 numbers now under our consideration. 



Prof. Sylvester was the editor-in-chief until his return 

 to this country ; now the mantle has fallen upon his suc- 

 cessor, Prof. Xewcomb, under whose auspices vol. vii. is 

 being published. Dr. Thomas Craig has been the 

 assistant editor during the issue of all the numbers. 



The chief papers treat of the higher algebra. In this 

 branch the contributions of Prof. Sylvester naturally loom 

 large. They are " On Sub- Invariants, i.e. Semi-Invariants 

 10 Binary Quantics of an Unlimited Order," " Tables of 

 Generating Functions, reduced and representative for 

 certain Ternary Systems of Binary Forms" (the "Tables" 

 were calculated by Messrs. Durfee and Ely), " A Con- 

 structive Theory of Partitions, arranged in Three Acts, 

 an Interact, and an Exodion," a most valuable contribu- 

 tion to the theory, written with the author's characteristic 

 fervour, but perhaps the gem of the collection is the first 

 instalment of the "Lectures on the Principles of Uni- 

 versal Algebra." 



We naturally turn next to the papers by Prof. Cayley. 

 These are a " Xote on a Partition-Series," " A Memoir on 

 Seminvariants," following up a "remarkable" discovery 

 by Capt. Macmahon, which leads to the conclusion that 

 the theory of seminvariants is a part of that of symmetric 

 functions, and three sets of tables, viz. non-unitary 

 partition tables, seminvariant tables, and tables of the 

 symmetric functions of the roots, to the degree 10 for the 

 form — 

 I + 6x + CX*/l-2 + . . . = (I - ax) (I - /3.i) (I - yx) . . . 



Following in the wake of these leviathans, Mr. Durfee 

 contributes " Tables of the Symmetric Functions of the 

 Twelfthic," and "The Tabulation of Symmetric Func- 

 tions " ; Capt. Macmahon writes on " Seminvariants 

 and Symmetric Functions," "Symmetric Functions of the 

 i3 : ." and " 1 Mi Perpetuants " : he is also the author of a 

 short " Xote on the Development of an Algebraic Frac- 

 tion," the moving cause of which is a previous article by 

 M. Faa de Bruno, entitled " Sur le de'veloppement des 

 fonctions rationnelles," which in its turn owed its origin to 

 a note by Prof. Sylvester in the Johns Hopkins Circulars. 

 Mr. J. Hammond, another worker in this field, has a 

 paper " On the Solution of the Differential Equation of 

 Sources," in which he gives a disproof of Prof. Sylvester's 

 fundamental postulate, a discovery which he first com- 

 municated to the London Mathematical Society. Mr. G. 

 S. Ely applies the method of graphs to compound par- 

 titions, and Mr. Morgan Jenkins gives a proof of a 

 theorem in partitions, and furnishes a note on Prof. 

 Vol. xxxi. — No. 792 



Sylvester's constructive theory of partitions, mentioned 

 above. 



We pass from this group of subjects, which centres 

 more especially round the name of Sylvester, and come 

 to papers on elliptic functions in one or other of the 

 forms under which that branch is now ranged. M. Faa 

 de Bruno has a long article on " Quelques applications de 

 la theorie des formes binaires aux fonctions elliptiques " : 

 Dr. Craig contributes several papers, viz. " Some Elliptic 

 Function Formulae," " On a Theta-Function Formula," 

 " On Quadruple Theta-Functions " (two papers), " On 

 Theta-Functions with Complex Characteristics," and 

 "On Certain Groups of Relations satisfied by the Quad- 

 ruple Theta-Functions." Prof. W. W. Johnson presents 

 a proof of the imaginary period in elliptic functions ; Mr. 

 A. L Daniels communicates three notes on Weierstrass's 

 methods in the theory of these functions ; and Prof. 

 Cayley, in a memoir on the abelian and theta functions, 

 reproduces, with additional developments, the course of 

 lectures which he delivered at the Johns Hopkins Uni- 

 versity in the early months of 1SS2. 



The other papers on algebraical subjects may be 

 grouped together. They are :—" On Division of Series," 

 by Rev. J. Hagen ; " Tables for Facilitating the Deter- 

 mination of Empirical Formulae," by A. W. Hale; "On 

 the Development of an Algebraic Fraction," by Dr. 

 Franklin ; some papers " On the Theory of Xumbers," 

 by A. S. Hathaway; "Sur une formule relative a hi 

 theorie des fonctions d'une variable," by M. Hermite : 

 " Calculus of Direction and Position," by E. W. Hyde ; 

 "Compound Determinants," by C. A. Van Velzer (written 

 before the author had seen Mr. R. F. Scott's paper in 

 vol. xiv. of the London Mathematical Society's Proceed- 

 ings), in which is discussed Picquet's proof of a theorem 

 of Sylvester's. Mr. McClintock writes on the resolutions 

 of equations of the fifth degree, a subject which is also 

 handled by Mr. G. P. Young, who in addition discusses 

 the principles of the solution of equations of the higher 

 degrees. Mr. G. S. Ely furnishes some notes on the 

 numbers of Bernouilli and Euler (adopting a name given 

 by Sylvester), and gives a useful bibliography of Ber 

 nouilli's numbers. Such lists as these are of great service 

 to workers. 



Dr. Story defines the absolute classification of loci to 

 be that classification which is not altered by any real 

 linear transformation, and which is identical with the 

 ordinary classification in so far as the latter is inde- 

 pendent of all consideration of the nature of the infinite 

 elements of the loci ; a part of this classification has been 

 made (as Dr. Story remarks) in essence by Prof. Sylvester 

 in the Phil. Mag. (February 1851)- The title of the 

 paper is " On the Absolute Classification of Quadratic 

 Loci, and on their Intersections with each other and with 

 Linear Loci." The same author also contributes t w < > 

 articles on the non-Euclidian geometry : one is a con- 

 tinuation of a paper by him in vol. iv., and in it are given 

 a number of formukc relating to distances, angles, area-. 

 and volumes ; the other is entitled " Non- Euclidian Pro- 

 perties of Conies," and contains an application of Prof. 

 Cayley 's projective measurement, generalised by Klein, 

 and still further extended by the author in the paper 

 just cited, to the conic. 



Dr. Franklin discusses some points in the theory of 



