February 13, 1902] 



NA TURE 



357 



fection in material submitted for examination. In one case, 

 plague infection was found to be undoubtedly present, and the 

 report of the laboratory upon it was made immediately. In a 

 second case the appearances presented were suspicious, and the 

 report of the laboratory enabled precautions to be taken to safe- 

 guard the city in the event of true plague appearing. More 

 accommodation for this Uind of research work is required, as 

 there is no lack of persons willing to undertake it. The same 

 remark applies to other departments of the College. If people 

 of means in Bristol and the neighbourhood took any interest in 

 educational progress, the establishment of the University of 

 Birmingham, and the movement in favour of other universities 

 of a similar kind, would inspire them to action in the same 

 direction. There is room for a University of the West of Eng- 

 land, and if Bristol does not rise to its opportunity another city 

 of the west will take its place. The subject has been brought 

 up over and over again, and only a few days ago Mr. Haldane 

 spoke in favour of it at the annual dinner of the University 

 College Colston Society. The Bishop of Hereford also alludes 

 to it at the end of the present report. But the rich citizens of 

 Bristol do not seem to understand what has been done by private 

 persons for higher education in cities like Liverpool, ISirming- 

 ham and Manchester, or if they know they apparently have no 

 desire to follow the example. There will have to be a complete 

 awakening of the spirit of pride in local resources for education 

 and research before Bristol can make any real movement towards 

 a University of the West of England. 



SCIENTIFIC SERIALS. 



Transactions of the Amcrkan Mathematical Society, vol. iii" 

 No. I, January. — On a class of automorphic functions, by J. L. 

 Hutchinson. In Burkhardt's " Ueber die darstellung einiger 

 falle der automorphen primformen durch SpecielleThetareihen," 

 the following monodromy group of the Riemann surface, y^ = 

 (i- - a,) (j:- 02) (jr- 6[)' {.< - S,,)-, is considered, and he shows 

 how a certain prime form which is automorphic for the group 

 can be expressed by a theta series. Further results are here 

 given concerning the group and the functions belonging to it, 

 the chief object being to obtain explicit analytic formulae by 

 means of which all functions of the group can be represented. 

 To this end the theta-fuchsian functions of Poincare are intro- 

 duced, and their expressions in terms of the hyperelliptic theta 

 series deduced. — Concerning the existence of surfaces capable 

 of conformal representation upon the plane in such a manner 

 that geodetic lines are represented by a prescribed system of 

 curves, by H. F. .Stecker, is in continuation of a previous paper 

 under nearly the same title (vol. ii. p. 152). — Zur erklarung der 

 Bogenlange und der inhaltes einer Krummen flache, by O. 

 Stoll(i/! the author's " Grundziige der Differential- und Integral- 

 rechnung,'' Bd. 2, and Malh. Ann., Bd. 18). — The groups of 

 Steiner in problems of contact, by Dr. L. E. Dickson, gives an 

 elementary proof of Jordan's ("Traite," pp. 229-249). Refer- 

 ence is given to Steiner and Hesse (Journal fur Afat/i., vol. 

 xlix. (1855) and vol. Ixiii. (1864), and to papers by the author 

 (Bulletin Amer. Math. Soc, vol. iv. , and the American 

 fournal of Mithematics, vol. xxiii. pp. 337-377). — (Quaternion 

 space, by A. S. Hathaway, follows up Stringham's work in vol. 

 ii. p. 183, but frequent reference is made to Clifford's paper on 

 biquaternions. Stringham deals analytically with the equations 

 of loci and develops the geometry by the interpretation of those 

 equations ; the author uses a more synthetic method, which 

 interprets the quaternion symbols themselves instead of the 

 equations between them. It is this divergence which con- 

 stitutes the general difference between the methods of Cayley 

 and Tait. Cliff ird slated the synthetic view in his Further note 

 on biquaierni'ins. — Reciprocal systems of linear differential 

 equations, by E. J. Wilczynski, arrives at interesting results in 

 connection with previous papers (Transactions, vol. ii. No. 4 ; 

 American Journal of Mathematics, vol. xxiii.). — On the in- 

 variants of quadratic differential forms, by C. N. Haskins, 

 investigates, by means of Lie's theory of continuous groups, the 

 problem of determining the number of invariants of the general 

 quadratic form in n variables. Numerous references occur in 

 the paper. — On the nature and use of the functions employed in 

 the recognition nf quadratic residues, by Dr. E. McClintock, refers 

 to Tannery, " Lemons d'Arithmetique," Bachmann, " Elemente 

 der Zahlentheorie," and to Baurngart, " Ueber der Quadratische 

 Reciprociiatsgezetz. " — k. determination of the number of real 



and imaginary roots of the hypergeometric series, by E. B. 

 Van Vleck. Concisely we must refer to Klein (Math. .Ann., 

 vol. xxxvii. p. 573) for the number of the roots of the equation 

 considered between o and - 00. Mr. Van Vleck claims that he 

 gives, for the first time, the number of imaginary roots. 

 Numerous references and diagrams (six and a page of sixteen) 

 accompany the text. — The second variation of a definite 

 integral when one end-point is variable, by G. A. Bliss. The 

 method which the author applies to the discussion of the case 

 in which one end-point moves on a fixed curve is closely 

 analogous to that of Weierstrass (" Lectures on the Calculus of 

 Variations," 1879). In the present case terms outside of the 

 integral sign are taken into consideration. Then, as a result of 

 the discussion, the analogue of Jacobi's criterion is derived, 

 defining, apparently in a new way, the critical point (Kneser's 

 " Brennpunkt ") for the fixed curve along which the end-point 

 varies. Then the relation between the critical and conjugate 

 points is discussed.— On the projective axioms of geometry, by 

 E. H. Moore, contains a consideration of the axioms called by 

 Hilbert ("Grundlagen der Geometrie") the axioms of connection 

 and of order, and by Schur (" Uber die Grundlagen der Geo- 

 metrie") the projective a.xioms of geometry. There are several 

 citations of authorities, such as Peano, Pasch and Ingrami. 



Bulletin of the .American Mathematical Society, ]3.nwi.iy.~ 

 Note on Mr. George Peirce's approximate construction for jt, by 

 E. Lemoine. This article gives four constructions suggested by 

 a discussion of Mr. Peirce's which we have previously noticed 

 (Bulletin, ]u\y, 1901). The relative theoretic exactness is de- 

 termined by calculating the true value of the length which in 

 each case approximately represents ir. The solutions are 

 worked out by aid of the geometrographic notation. A slight 

 sketch of this method, sufficient for the present purpose, is given. 

 (For a fuller account, reference may be made to M. Lemoine's 

 " La Geomerografie " or to the "Traite de Geometrie" in the 

 .Archiv. der Mathematik und Physik, April and May, 1901, 

 vol. i., Gauthier-Villars). There is also appended a close 

 approximation to the trisection of an angle by C. Stormer 

 worked by the same method. There are several diagrams. —Con- 

 cerning the elliptic ^ (*o,,?|j,-) functions as coordinates in aline 

 complex, and certain related theorems, by Dr. H. F. Stecker, is an 

 application of the coordinates to the Kumnier surface and certain 

 other configurations (cf Klein, Math. Ann. vol. v., pp. 294-5). 

 —A short note on the Abelian groups which are conformal with 

 non-Abelian groups follows, by Dr. G. A. Miller. Dr. S. E. 

 Slocum writes on the infinitesimal generators of certain para- 

 meter groups. The paper opens with a nlsumJ of the method 

 employed by the author on pp. ()^-la->,o^i'a& Proceedings of 

 the American Academy of Arts and Sciences, vol. xxxvi., and 

 then proceeds to give tables in which are enumerated all pos- 

 sible types of structure of two-, three- and four-parameter com- 

 plex groups as given by Lie, and under each structure are given 

 the symbols of the infinitesimal transformations which gen- 

 erate the parameter group corresponding to that structure, 

 obtained by the method relerred to above (r/ Lie, " Continuier- 

 liche Gruppen," pp. 565-589 ; "Transformations Gruppen," 

 vol. iii., pp.713-730).— Notices follow of the " Einfiihrungindie 

 Theorie der Differential gleichungen mit einer unabhangigen 

 Variabeln" of Dr. L Schlesinger and of Prof. Hatzidakis's 

 "■ iiaayaiyr\ eis t^v 'kuanefav ' AKyeSpaf." — The usual informa- 

 tion of interest to mathematicians follows in the form of notes 

 and new publications. 



Annals of Mathematics, January.— Some applications of 

 the method of abridged notation, by Maxime Bocher, is an 

 interesting elementary paper the nature of which may be 

 gathered from an illustration. Let the sides of a triangle be 

 « = o, z' = o, ?o = o (where ?<=.vcos a + ysina-p), then u-v = q, 

 v-w = o, m- u = o Ate ihe bisectors of the angles, and as the 

 sum of the sinisters vanishes identically we get the property 

 of the bisectors of the angles cointersecting in a point. If, 

 again, u^.x-+y" + ax+ t>y +c = o and so on, we can show that 

 the common chords of three circles meet in a point. The author 

 then proceeds to the proof of Desargues' theorem and thence to 

 generalisations for four circles and for curves of the «th order, and 

 extends, by suggestion, his results to surfaces. — On the roots of 

 functions connected by a linear recurrent relation of the second 

 order, by M. B. Porter, reproduces in part some unpublished 

 theorems of Sturm (cf " Liouville," vol. i.), and shows how, 

 by means of the Cauchy-Lipschitz theorem for the existence of 

 solutions of a differential equation, it is possible to establish 



NO. 1685, VOL. 65] 



