April II, 18S9] 



NATURE 



571 



approximation to the true theoretical shape of the arch is 

 attained either by taking the "three-point circle," passing 

 through the vertex A and the two points of equal curva- 

 ture Bj and B,', or by taking the "described circle," touching 

 at Bj and B/, or by taking the "inscribed circle," touching at 

 A and again internally at B3 and B3' beyond the points of 

 maximum curvature Bj and Bj', instead of taking, as customary, 

 the circle of curvature at the vertex ; and the authors show that 

 if an elliptical arch is described, the proper approximation to its 

 shape is obtained from an orthogonal projection of one of these 

 circles. 



The points Bj and B/ are called the noses of the transformed 

 catenary, and give the name to the paper. 



The transformed catenary, which may be taken 'as the line of 

 thrust, is shown to lie below the "described circle," and above 

 the " three- point circle," so that by taking these or similar 

 circles for the boundaries of the ring of the arch, the proper 

 stability of the arch is secured. 



The mathematical treatment of the Catenary given by the 

 authors would gain considerably in elegance by the employment 

 of the hyperbolic functions, now no longer to be disregarded ; 

 thus, instead of writing — 





-f ^ 



the notation — 



tan = -^^' -_/:(, 



^ = r cosh 5, tan 9 = r sinh -, 

 m m m 



should be employed ; and for purposes of numerical calculation 

 of these hyperbolic functions, it is only necessary to notice that 

 if tan e = sinh u, sec = cosh u, then u = hyp. log (sec 6 + 

 tan 6) ; so that the table of u, already calculated by Euler, used 

 in conjunction with the tables of the ordinary circular functions, 

 will give us the numerical values of the hyperbolic fnnctions ; 

 for large values of u, when $ denotes an angle being nearly a 

 right angle, the approximate equation — 



log cosh u — log sinh u = ulog e - log 2 

 being sufficient. 



Tables of numerical results are eiven at the end of the paper, 

 with practical illustrations, for the benefit of architects and 

 engineers, and supplementary tables are added for the imme- 

 diate designing of brick, sandstone, and granite arches, with 

 circular soffits ; so that these investigations should prove 

 decidedly useful to those engaged in the design of similar 

 structures. A. G. G. 



SCIENTIFIC SERIALS. 



American Journal of Mathematics, vol. xi. No. 3 (Baltimore, 

 April 1889.)— Were it not for the size of the pages, this number 

 might be taken to be a number oi ihe Maiheniaiiscke Annalen, 

 seeing that out of its ninety-eight pages sixty-eight are written in 

 German. The first memoir, by Oskar Bolza (pp. 195-214), is 

 entitled "On the Construction of Intransitive Groups," and 

 touches on points discussed by Jordan (" Traite des Substitu- 

 tions"), Capelli ("vSopra risomorfismo dei Gruppi di Sostitu- 

 zioni "), Netto (" Substitutionentheorie "), Cayley (" Theory of 

 Groups"), and Dyck ("Gruppentheoretische Studien "). This 

 is followed by a short note by Karl Heun (pp. 215-20), on 

 " Die herstellung einer linearen Differentialgleichung aus 

 einem gegebenen Element der Integralfunction." Next we 

 have an important memoir by Koenigsberger (op. 221-82), 

 " Ueber die Reduction von Inlegralen transcendenter Func- 

 tionen." The closing note, by Dr. Franklin (pp. 283-92), 

 "On the Double Periodicity of the Elliptic Functions," m/r;- 

 alia, proves a theorem of a bicircular quartic, enunciated by 

 Clifford (see Crofton, L.M.Soc Proc, vol. ii), and also results 

 due to Siebeck and Greenhill, but all are established here from 

 a different point of view. 



Kivisla Scientifico-Indiistriale, February 15. — Researches on 

 the thermo-electric conductivity of magnetized iron (concluded), 

 by Prof. Ercole Fossati. From these experiments the author 

 concludes generally that the electric conductivity of iron either 

 suffers no change under transverse magnetization or undergoes 



some increase ; this increase, however, is much less than the 

 diminished conductivity of iron magnetized in the longitudinal 

 direction. This inference agrees perfectly with the deductions 

 he had already arrived at experimenting with iron conductors of 

 varying dimensions. — Some experiments with a new battery, by 

 Prof. Augusto Righi. Excellent results have been obtained from 

 the battery here described, which consists of 108 condensers dis- 

 posed in six groups of eighteen each, one above the other, in 

 order to obtain high potentials. The outer armatures com- 

 municate with the conductors of a Holtz machine, the central 

 with the ground, and a capacitity is thus obtained equal to that 



of ^- = 3 jars, united in a single battery with armatures com- 

 municating directly with the two conductors of the machine. 

 The capacity of the whole system is thus 18,810 electrostatic 

 units, or about ,V microfarad. 



Rendiconti del Reale Istituto Lombardo, February 28. — Notice 

 of the late Prof. Giuseppe Meneghini, by Prof. T. Taramelli. 

 In this biographic memoir a summary account is given of the 

 services rendered to geological studies by the eminent naturalist, 

 who was born in Padua in i8ii, and died in January of the 

 present year. Meneghini is best known as joint translator, with 

 Savi, of Murchison's work on the Alps, Apennines, and Car- 

 pathians, and by his systematic monograph on the fossils of the 

 Upper Lias in Lombardy and Central Italy. His last publica- 

 tion was a paper on the Cambrian tnlobiles of the Iglesiente 

 district, and his name will always be associated rather with the 

 palaeontological than the stratigraphic or petrographic side of 

 geology. — Meteorological observations made at the Royal Brera 

 Observatory, Milan, during the month of February. 



The last issue of the Memoirs of the St. Petersburg Society of 

 Naturalists (vol. xix.. Botany) contain*, besides several very 

 interesting short papers in the Proceedings, a new contribution 

 to the flora of Novgorod, by A. Antonoff, which raises the 

 number of species of flowering plants discovered in the Govern- 

 ment of Novgorod to 700 ; a note on the comparative anatomy 

 of the tissues in the leaves of .Salicinecv as a basis for classifica- 

 tion, by V. Dobrovlanski ; and a suggestive description of 

 the flora of the Shenkursk and Kholmogory districts of Archan- 

 gelsk, by N. Kuznetsoff. Owing to the extension of a subsoil 

 of limestone, which is much warmed by the sun during !he 

 summer, the flora of the region contains a number of species 

 belonging to a more congenial climate, while several species 

 characteristic of those latitudes are wanting. On the other 

 hand, owing to its proximity to the Urals, and the existence on 

 the west of such a barrier as Lake Onega, the flora contains 

 a considerable number of Siberian species, while many West 

 European species do not appear. M. Kuznetsoff's remarks on the 

 extension of the Abies sibirica, which is slowly advancing towards 

 the West, and the lime-tree, which seems to be, on the contrary, 

 disappearing, are very interesting. Both the Ulmiis cainpestris 

 and the U. effusa were found as far north as the sixty-third 

 degree of latitude. The presence of the Lymnanthemum nymph- 

 oides, which is found in Lithuania and South -Eastern Russia, 

 but not in Central Russia, save Kursk, is especially remarkable, 

 and M. Kuznetsoff explains the extension of this aquatic plant 

 over Archangelsk by its having spread along the canal which 

 connects the Volga with the Dvina. No fewer than twelve other 

 South Russian plants, which must have migrated along the 

 same canal, are named by the author. As to the Mtdgcdium 

 tataricum, C. A. Mey., which is characteristic of the salt 

 steppes of Astrakhan, it has been found on a shoal of the Dvina, 

 under 64° N. lat. The paper is accompanied by a map, showing 

 the western limits of extension of the Abies sibirica. 



SOCIETIES AND ACADEMIES. 

 London. 



Royal Society, March 14. — "On the Organization of the 

 Fossil Plants of the Coal- Measures. Part xvi." By W. C. 

 Williamson, LL.D., F. R.S., Professor of Botany in the Owens 

 College, Manchester. 



In this memoir the author first calls attention to detached 

 observations, made in his earlier memoirs, relating to the manner 

 in which a medullary axis is developed in the interior of each of 

 the primary vascular bundles of the Carboniferous Lycopodi- 

 aceae. He then traces the changes undei^one dur.ii.r the deve- 

 lopment of a small branch-bundle in Lepidodendron Harcourtii. 



