October 4, 1 900] 



NATURE 



56! 



spondence College Press: — "Algebra, The Tutorial, Part I., 

 Klementary Course," by Rupert Deakin ; "Arithmetic, The 

 Tutorial," by W. P. Workman ; " Building Construction (Science 

 and Art)," by Brysson Cunningham ; " Machine Construction, 

 I'irst Stage (Science and Art)," by J. Handsley Dales ; " Ma- 

 ihematics. First Stage (Science and Art)" ; " Physiography, 

 Section One (Science and Art)," by Fabian Rosenberg ; " Prac- 

 tical Plane and Solid Geometry, First Stage (Science and Art)," 

 by G. F. Burn. 



Mr. T. Fisher Unwin will add to his " Masters of Medicine" 

 Series, "Thomas Sydenham," by J. F. Payne, and "Andreas 

 Vesalius," by C. L. Taylor. 



Messrs. Frederick Warneand Co. will issue new editions of: — 

 " The Cattle D ctor," by Geo. Armatage ; " Wayside and 

 Woodland Blossoms, First and Second Series," by Edward Step. 



Messrs. Wells Gardner, Darton and Co.'s list includes a new 

 edition of" Playing at Botany," by Phoebe Allen. 



Messrs. Whittakerand Co.'s announcements are : — " Periodic 

 Classification and the Problem of Chemical Evolution," by 

 George Rudorf; "Inspection of Railway Material," by G. R. 

 Bodmer ; "Electric Wiring Tables," by W. Perren Maycock ; 

 "Telephone System of the British Post Office," by T. E. 

 Herbert; and "Horseless Road Locomotion," by A. R. 

 Sennett. 



MA THE MA TICS A T THE BRITISH 

 ASSOCIA TION. 



'PHE mathematical communications to this year's meeting of 

 the British Association were made on Monday, September 

 10, in one of the halls assigned to the Mathematical-Physical- 

 Astronomical Section. Major P. A. MacMahon, F.R.S., took 

 the chair. 



The committee appointed in 1888 to calculate tables of certain 

 mathematical functions opened the proceedings by presenting a 

 report on their year's progress. The work on which they have 

 for some time been engaged, namely, the preparation of a new 

 " Canon Arithmeticus," is now almost completed. The calcu- 

 lations have been made by Lieut. -Colonel Allan Cunningham, 

 who, in presenting the report, announced that the liberality of 

 the British Association and of the Royal Society had enabled 

 the committee to undertake the publication of the tables as a 

 separate volume. Before the Association meets next year this 

 will probably have been given to the world, and the committee, 

 after an existence of thirteen years, will (unless some other work 

 is found for it) cease to exist. 



Another report was taken next — this time not of a committee, 

 but of a single worker. Miss F. Hardcastle, of Cambridge, who 

 was commissioned two years ago to prepare an account of " The 

 present state of the theory of point-groups " for the Association. 

 In the absence of Miss Hardcastle, one of the secretaries stated 

 that a first instalment of the work is to be published in this 

 year's annual report ; this, however, will give only the general 

 classification of the subject, and an account of those memoirs on 

 the theory of elimination which are of importance in it. The 

 greater part of Miss Hardcastle's report will not be ready until 

 next year. 



The chair was then taken by Prof. Forsyth, while Major 

 MacMahon read a paper on " A property of the characteristic 

 symbolic determinant of any « quantics in n variables." Let 



{1 h in 



a\x, a<lxy • • • «nx 



-J (in symbolic notation) any « quantics in m variables, and let 





+ C 



h (2 



ii €2 



in 



X„+ . . 



Major MacMahon arrives at the remarkable result that 



(where the summation is extended over all positive integral values 



of (1, fa. • • • I") has the vali 





where / (0) is the 



characteristic determinant of the umbrse a,„ a^.^ 

 NO. I 6 14, VOL. 62] 



The next communication was made in French by Prof. 

 Cyparissos Stephanos, of the University of Athens, " Sur les 

 relations entre la geometric projective et la mecanique." The 

 fundamental thought of this paper may be explained as follows. 

 Consider a system of forces in equilibrium. What geometrical 

 transformations of space will transform this system into another 

 system of forces also in equilibrium ? Prof. Stephanos solves 

 this problem, and finds that the only transformations which 

 thus conserve equilibrium are those which, considered as per- 

 formed on the Pluckerian co-ordinates of the forces, are linear and 

 homogeneous. When the system of forces is coplanar. these trans- 

 formations are homographies in the plane. This train of thought 

 is of some importance in Graphical Statics. 



Mr. H. S. Carslaw (Fellow of Emmanuel College, Cambridge) 

 followed with a paper on " The use of multiple space in applied 

 mathematics." The method of images, so powerful in electro- 

 statical problems, can in its original form be applied only when 

 the fundamental angles of the problem are submultiples of v. 

 Prof. Sommerfeld pointed out a year or two ago that by intro- 

 ducing the idea of a branched space, analogous to the branched 

 planes used in Riemann's Theory of Functions, the method of 

 images can be freed from this limitation. Mr. Carslaw's work 

 is an extension and development of this suggestion, which is 

 applied by him to the solution of several of the standard problems 

 of the potential theory. 



Lieut. -Colonel Cunningham then gave some results obtained 

 by himself and Mr. H. J. Woodall in the "Determination of 

 successive high primes." As an Example of a new process due 

 to the authors, the factors of all numbers between 16 776 196, 

 and 16 778 236 have been determined. 117 of the numbers 

 in this series are found to be primes, a fact which led to some 

 discussion on Riemann's work in the theory of prime numbers. 



This was followed by a paper on "The construction of magic 

 squares," by Dr. J. Willis, of Bradford, in which some new 

 modes of formation were described and exemplified in diagrams. 

 Major MacMahon then communicated two papers in succession. 

 The first was entitled "The asyzygetic and perpetuant 

 covariants of systems of binary quantics"; it was concerned 

 with the extension, to a system containing any number of binary 

 quantics, of the work which has already been done in connection 

 with the semivariant forms of a single binary quantic. 



In the second paper, " On the symbolism appropriate to the 

 study of orthogonal and Boolian invariant systems which apper- 

 tain to binary and other quantics," Major MacMahon explained a 

 new and most remarkable method which he has discovered in 

 the invariant theory, which promises to revolutionise the treat- 

 ment of that subject. Previous writers have considered the 

 invariant theory as consisting in the investigation of those 

 forms associated with a quantic, which are invariant when the 

 variables of the quantic are subjected to the general linear 

 transformation. When the variables are subjected only to 

 linear transformations of special types, such as the orthogonal 

 and Boolian transformations, the family of invariant forms 

 associated with a given quantic is, of course, much larger ; but 

 these special classes of transformations have hitherto been, 

 comparatively speaking, ignored, as forming a tedious and out- 

 lying branch of the subject. Major MacMahon's discovery is 

 a new symbolic method for obtaining the forms which are 

 invariant for orthogonal and Boolian transformations, in the 

 same way as Aronhold's symbolic method enables the investigator 

 to obtain the forms which are invariant for the general linear 

 transformation. Major MacMahon obtains six symbolic factors 

 analogous to Aronhold's symbolic factors a^- and {a/>), and the 

 ordinary invariant-theory can be derived as a particular case of 

 the new theory, by simply rejecting those forms which contain 

 any one of a certain four of these factors. 



A paper by Mr. A. B. Basset, F.R.S., in which the result 

 that "aquintic curve cannot have more than 15 real points 

 of inflexion " — an extension of a theorem of Zeuthen's on 

 quartic curves — is obtained, was briefly communicated by the 

 chairman ; and a remarkably interesting session closed with two 

 communications by Prof. J. D. Everett, F.R.S., "On Newton's 

 contributions to central-difference interpolation," and " On a 

 central-difference interpolation formula." In the former of 

 these papers the author observed that certain formulce in the 

 calculus of finite differences, usually attributed to Stirling, 

 were really known to Newton ; in the second, a formula of in- 

 terpolation was obtained which is less unsymmetrical than those 

 generally given. E. T. Whittaker. 



