May 4, 1893] 



NATURE 



graphs, associated the name of CliflTord only with the graphs in 

 the title. He had, however, on further consideration come to 

 the conclusion that by such exclusive association an impression 

 might be created which would operate unjustly towards the 

 unquestionable originality of the paper by Prof. Sylvester on 

 the application of the new atomic theory to the graphical repre- 

 sentation of the invariants and covariants of binary quantics 

 published in the American jfotirnal of Mathetnatics, vol. i. 

 p. 64. By permission of the council he proposed therefore to 

 alter the title of his paper by referring therein to the graphs as 

 "the Sylvester-Clifford graphs." — The following communica- 

 tions were made: — Toroidal functions, by A. B. Basset, F.R.S. 

 The object of this paper is to develop the theory of toroidal 

 functions from a point of view which brings out its connection 

 with the associated functions 



n It 



which occur in spherical and spheroidal harmonic analysis. A 

 toroidal function is an associated function of degree «— J and 

 order m, where n is zero or any positive integer, and «/ is zero 

 or any positive integer not greater than «. The paper com- 

 mences by showing that these functions may be expressed in 

 terms of the definite integrals — 



P"V) 



_(- 



)"'r(« + 4) 

 m + i 



and 



q;:>) 



_(-)"'r(«-)-4) 



/IT cos m<l>d<p 

 (,{•> + (y'-' - I)* cos i; 



cosh mipiitp 



/. 



V(ii-m + \) j^{v + {_i^-l)icosii<p\ 



,« + 4. 



n+\. 



It can easily be proved that these definite integrals satisfy the 

 differential equations for toroidal functions, and the advantages 

 of this method of procedure are twofold. In the first place 

 these integrals lead to certain difference and mixed difference 

 equations connecting functions of different orders and degrees ; 

 and in the second place the whole of the analysis and the results 

 will apply when « is changed into « + i, in which case the in- 

 tegrals become ordinary associated functions. In physical in- 

 vestigations connected with circular vortex rings, functions of 

 degree k and order unity occur, whose properties may be more 

 simply deduced from those of the zonal functions ; also i/ = cosh 77, 

 when 7) is very large. If, therefore, ('''=k, k will be small, 

 and appropriate series can be obtained in terms of /•. The 

 latter part of the paper is occupied with the investigation of 

 these series, and it is shown that 



whilst 



P„ = ,J-''+!(,^„ log4/^-Hf„), 



where (/>«, -i/n are infinite series of powers of k"^. — Note on the 

 problem to inscribe in one of two given triangles a triangle 

 similar to the other, by Mr. J. Griffiths. The writer discusses 

 the following propositions: (l) A triangle DEF inscribed in a 

 given triangle ABC, so as to be similar to another given one 

 A'B'C, belongs to some one of twelve systems of similar in- 

 triangles, each system having a centre of similitude of its own. 

 (2) The centres of similitude of the twelve systems in question 

 can be formed into two groups of six points which lie respec- 

 tively on two circles, inverse to each other with respect to the 

 circumcircle ABC. (3) The centre of similitude of any system 

 of similar triangles inscribed in ABC and having a common 

 ■Brocard angle equal to that of A'B'C will lie on one or other of 

 the above circles. (4) As a particular case of the problem the 

 different systems formed by a triangle DEF inscribed in ABC, 

 so as to be either directly or inversely similar to it are noticed. 

 — The singularity of the optical wave-surface, by J. Larmor, 

 F.R.S. It is shown that two sheets of a wave-surface cannot 

 intersect along a curve. As the elastic quality of a crystalline 

 medium is gradually altered, the separate sheets of its (me- 

 chanical) wave-surface may draw together, and may finally come 

 into contact at one or more conical points ; but any further 

 alteration in the same direction produces instability. The 

 existence of the abnormality of conical refraction would thus be, 

 on a purely elastic theory, an indication of the immediate 

 approach of instability. — On a problem of conformal representa- 

 tion, by Prof. W. Burnside. The paper deals with those cases 

 in which a rectangular polygon can be represented conformally 

 on a circle or half plane by means of an integral equation 

 Ijctween two complex variables. It is formally proved that 



NO. 1227, VOL. 48] 



whenever the polygon can be formed by the juxtaposition of 

 equal and similar figures either 



( 1 ) triangle ~.~-y 

 236 



(ii) triangle 



442 



(iii) any rectangle 



the representation is possible by such an integral equation, and 

 that it is not possible in any other case. A general method for 

 finding the equation carrying out the representation is given, 

 and a few special cases are worked at length. The paper finishes 

 by considering shortly the case in which the polygon is not 

 simply connected, and one or two other allied points. 



Linnean Society, April 20. — Prof. Stewart, President, in 

 the chair. — In view of the approaching anniversary meeting the 

 election of auditors took place, when Dr. Meiklejohn and Mr. 

 E. A. Batters were nominated on behalf of the Council, and 

 Messrs. Thomas Christy and W. F. Kirby on behalf of the 

 Fellows. — The President took occasion to notice the retirement 

 of Mr. F. H. Kingston after thirty-six years' service as lodge- 

 keeper, and presented him with a testimonial in the shape of a 

 cigar case containing five and thirty pounds in bank-notes, which 

 hadbeen subscribed on his behalf by all thesocieties in Burlington 

 House. After a suitable response on the part of the recipient, 

 attention was directed to some photographs of Burlington House 

 with the gateway as it existed before the rebuilding in 1868, 

 and showing the old colonnade which had since been demolished 

 and was lying still uncared for in Batlersea Park. — On behalf of 

 Mr. C. Chilton of Dunedin, N.Z., Mr. W. Perry Sladen gave 

 an abstract of a paper on the subterranean Crustacea of New 

 Zealand, with remarks on the fauna of caves and wells. The 

 paper contained a raww/ of previous publications on the subject 

 with additional information from the author's own observation, 

 and an expression of his views on certain controversial points in 

 connection therewith. His remarks were criticised by the 

 President and by Prof. Howes, Dr. Henry Power and Mr. G. 

 Fookes. — A paper was then read by Mr. H. M. Bernard on the 

 anatomy, physiology, and histology of the Chernetida, withspecial 

 reference to the rudimentary stigmata, and to a new form of 

 trachea, on which an interesting discussion ensued, and Mr. 

 Bernard replied to the criticisms which were offered. — The 

 society adjourned to May 4. 



Paris. 

 Academy of Sciences, April 24. — M. Loewy in the 

 chair. — On the observation of the partial solar eclipse of April 

 16, made at the Paris Observatory, by M. F. Tisserand. From 

 a measurement of six photographs obtained by MM. Henry, 

 the instants of contact were calculated to have been 3h. S9ni. 51s. 

 and 4h. 27m. 59s. — Recent researches on the nitrogen-fixing 

 micro-organisms, by M. Berthelot. From a series of experi- 

 ments upon samples of earth taken from the Botanic Garden of 

 the Ecole de Pharmacie, it appears that the micro-organisms 

 capable of fixing free nitrogen from the air belong to widely 

 varying species, but that the chief agents are certain bacteria of 

 the soil, seven species of which were isolated. The carbon and 

 hydrogen contained in the atmosphere does not appear capable 

 of supporting the life of these bacteria, and their nourishment is 

 chiefly derived from the decomposition of sugar, tartaric acid, 

 and other hydrocarbons supplied by higher organisms. If 

 there is an abundance of combined nitrogen at hand, the bacteria 

 flourish more profusely, and their absorption of free nitrogen, 

 though placed beyond doubt, has certain definite limits. On 

 the whole, it seems that the carbon-fixing and the nitrogen- 

 fixing organisms fulfil mutually supplementary functions. — On 

 the order of successive appearance of the vessels in the parallel 

 formation of the leaves of certain Composite ( Tragopagon, &c.), 

 by M. A. Trccul. — Physiological and therapeutic effects of a 

 liquid extracted from the male sexual gland, by MM. Brown- 

 Sequard and d'Arsonval. Samples of the orchitic liquid for 

 subcutaneous injection were offered to all medical men willing 

 to report upon its effects. Over 1200 physicians availed them- 

 selves of this offer, and their results are very encouraging. The 

 malady showing the most striking effect of the remedy was loco- 

 motor ataxy, of which 314 out of 342 undoubted cases were 

 cured or considerably improved. Another almost incurable 

 disease which proved very amenable to this treatment was 

 shaking paralysis, of which 25 out of 27 cases were much im- 

 proved. It appears that the orchitic liquid, though not possess- 



