Februaby 1G, 



sciencje;. 



43 



tends the theory of Stehier's polygons and Prof. 

 Sylvester's theory of derivation to the case of curves 

 of order n and deliciency p in an {n-p) flat. The 

 extension involves the use of Abelian functions 

 instead of elliptic functions, as in the case of a plane 

 cubic, and is based principally upon Clifford's well- 

 known paper, On the classification of loci. — {Proc. 

 Land. math, soc, 1SS2. ) T. c. [84 



Fourier's functions. — M. Nicolas prefers to de- 

 note, by this title, the functions more commonly 

 known as Bessel's or cylindric functions. The author 

 studies principally the different modes of representa- 

 tion of 'these functions by definite integrals and series. 

 A novelty is the introduction of a method of Euler's 

 in finding the development in form of a series of the 

 functions of the second kind. — {Annates ecole norm., 

 :^i., supjyl., 1S82.) T. c. [85 



Geometry of n-dimensions. — The author, M. 

 V. Schlegel, here extends certain well-known theo- 

 rems of ordinary plane and three-dimensional space 

 geometry to a space of any number of dimensions. 

 The paper deals only with completely limited figures, 

 regular and iiregular. A homorjeneously limited Jiynre 

 is defined: 1°, as one in which each summit meets . 

 the same number of edges, planes, solids, etc. ; 2°, as 

 one in following any edge of which we meet the same 

 number of edges, planes, etc. Wi'iting 'homogene- 

 ous ' instead oi 'limited homogeneously,' we see that 

 all plane polygons are homogeneous, etc. The author 

 uses the methods of Grassman, and extends to hyper- 

 space theorems concerning the triangle, quadrilateral, 

 tetrahedron, hexahedron, and octohedron. — (Bull, 

 soc. math. France, X., 1SS2.) T. c. [86 



Curves -whose co-ordinates are elliptic func- 

 tions. — R. von Lilienthal discusses two classes of 

 spherical curves having the following properties : The 

 conslants in the expressions for the co-ordinates, with 

 the exception of one (which, with the modulus, is 

 arbitr-ary), can be so determined that the sought curve 

 shall lie on a sphere. The length of an arc of the 

 curve can be given as an elliptic integral of the first 

 kind increased by the difference of two elliptic inte- 

 grals of the third kind. 



For the second group of curves, the arbitrary con- 

 stant can be so determined that the integral giving 

 the length of arc shall be an elliptic integral of the first 

 kind, "it is also shown that the curves of the second 

 kind lie on algebraical cylinders. — [Journ. reine an- 

 'jeic. mnUi., xciii., 1882.) T. c. [87 



Applications of the theory of binary forms to 

 elliptic functions. — The author, Fah, de Bruno, ex- 

 presses the elliptic functions by aid of the absolute in- 

 variant, and gives a very rapidly converging series for 

 the computation of the complete elliptic integral of 

 thefirstkind. — (^j)ie/-.jo!(r)i.)jia(/i.,lSS2.) T. c. [88 

 Rotation of a solid body. — This treats the case 

 of rotation of a solid body about a point which is in 

 genei'al neither the centre of gravity of the body nor 

 (in the case of a body of revolution) a point on the 

 axis of revolution. The author, W. Hess of Munich, 

 discusses the general case, and obtains several inter- 

 esting theorems on making particular hypotheses as 

 to the position of the point about which the body 

 rotates. — [ilaih. annalen-, xx., 18S2.) T. c. [89 



Vibrations of an elastic sphere. — Prof. H. 

 Lamb here discusses the problem of the vibrations of 

 an elastic solid whose dimensions are all finite. He 

 has given several numerical calculationsand diagrams, 

 illustrating, in special cases, the results arrived at by 

 the purely mathematical investigation. The author 

 points out that the results of his analysis differ from 

 the views advanced by Lame (Th^orie de I'^lasticite) 



as to the nature of the fundamental modes of vibra- 

 tion of elastic solids in general; and indicates the 

 error in Lam^'-s reasoning as consisting in the tacit 

 assumption that a wave undergoes no change of char- 

 acter on reflection at the bounding surface of a solid, 

 — an assumption the incorrectness of which was pre- 

 viously shown by Green. — (Proc. Lond. math, soc, 

 1882.) T. c. [90 



Subinvariauts. — An important paper by Professor 

 Sylvester, of which, since it is not yet completed, a 

 review will be given at a later date. — (Amer. journ. 

 mai/i., v., 1882.) T. c. [SI- 



PHYSICS. 

 Apparent attractions and repulsions of small ^ 

 floating bodies. — The need of a thoroughly soimd 

 and at the same time simple popular explanation 

 of capillary phenomena will probably make every 

 teacher of elementary physics take up Prof. Le- 

 conte's article with interest. As he states, ordinary 

 treatises are somewhat unsatisfactory upon this 

 subject, even when they are not actually wrong. 

 For instance, the in general excellent treatment of 

 capillary action in Everett's ' Deschanel ' handles the 

 phenomena observed in a vacuimi in a very gingerly 

 manner, hinting at a certain mysterious pressure in 

 the interior of liquids due to molecular action at the 

 surface, even when such surface is plane, in order 

 to account for the rise of liquids in fine tubes in a 

 vacuum. 



In view of the fact that the capillary action of 

 liquids is practically the same in a vacuum as in air, 

 Prof. Leconte appears to be of the opinion that it is 

 unnecessary to take account of atmospheric pressure 

 in explaining any of these phenomena. He proposes 

 to base his explanation iqwu two 'fundamental prin- 

 ciples :' 1. "That in every case, whether of moist- 

 enedvor non-moistened bodies, there exists an adhe- 

 sion between the solid and the liquid." 2. " That 

 the capillary forces are, in any given case, inversely 

 proportional to the radii of curvature of the menis- 

 cuses, and their resultants, directed toward the cen- 

 tres of concavity." 



We suppose Prof. Leconte will admit, however, that 

 although the visible phenomenon of water sustained 

 in a capillary tube, for instance, may remain un- 

 changed when the surface of the water is relieved of 

 the pressure of the atmosphere, the actual condition 

 of water in the tube and of the film at the top of the 

 column is somewhat changed. Thus Yoimg says in 

 his memoir on the 'Cohesion of fluids,' "when the 

 surface is concave, the tension is employed in coun- 

 teracting the pressure of the atmosphere, or, where 

 the atmosphei-e is excluded, the equivalent pressure 

 arising from the weight of the pai-ticles suspended 

 from it by means of their cohesion," etc. In fact, it 

 would seem the better plan in explaining the above 

 phenomenon, to make full use of the unquestionable 

 agency of the atmospheric pressure, so long as the 

 atmosijhere is present, and be thankful for it, since it 

 is far easier to understand than the sustaining by 

 cohesion that must take its place in a vacuum. 



Prof. Leconte's statement of Iiis second principle 

 is a little puzzling; for a natural interpretation of his 

 words would be, that he supposes the surface tension 

 to be inversely proportional to the radius of cur- 

 vature of the film. He applies his two principles to 

 the explanation of three typical cases of attraction and 

 repulsion. In the case of two moistened bodies he 

 says, " But when brought so near that their menis- 

 ouses join each other, the radius of curvature of the 

 united intervening concave meniscus ... is less 

 than that of the exterior concave meniscuses, . . . 



