4o6 



NATURE 



[February 23. 1893 



and describes their properties. — A method for the preparation 

 of acetylene, by M. W. Travers, Calcium carbide miy be pre- 

 pared by heatinc; a mixture of sodium, gas carbon, and calcium 

 chloride for half an hour at bright redness in an iron bottle. 

 The carbide 'bus obtained yields acetylene on treatment with 

 water. 240 c.c, half the theoretical quantity, of acetylene is 

 obtained for every gram of sodium used in the preparation of 

 the carbide. 



Mathematical Society, February 9. — Mr. A. B. Kempe, 

 F.R.S , President, in the chair. — The following communications 

 were made : — The Harmonics of a Ring, by Mr. W. D. Niven, 

 F.R.S. This paper treats of the potential functions of an anchor 

 ring, and explains how problems to which those functions are 

 applicable may be solved for two coaxal rings. The propositi >n 

 on which the method depends establishes that the ring har- 

 monics of any degree may be derived from their predecessors of 

 lower degree by simple differentiations with regard to the radius 

 of the dipolar circle of the ring and the distance of a fixed point 

 from the plane of this circle. Ultimately the haimonics depend 

 upon the potential at any point due to a distribution on the circle 

 either uniform or varying as a circular function of the arc. Now 

 the potential due to such distribution on a circle B may readily 

 be expressed in terms of the harmonics pertaining to a coaxal 

 circle A, and hence any harmonic pertaining to B, and therefore 

 any series of such harmonics, may be expressed in a series per- 

 taining to A. In the latter form they are suitable for the 

 application of surface conditions at any ring whose dipolar circle 

 is A. The application worked out in the paper is the problem 

 of the influence of two electrically charged coaxal rings upon 

 one another. It is also shown how the same problem may be 

 solved for a ring and sphere, symmetrically situated as regards 

 the axis. — The group of thirty cubes composed by six differently 

 ■coloured squares, by Major MacMahon, R.A., F.R.S. 

 Selecting any one of the thirty cubes at pleasure it is possible 

 to select eight of the remaining twenty-nine which in reference 

 to the cube selected have a very peculiar and interesting 



Selected CuDe (transformed) diagonal i6 vertica 



proJDerty. It is possible to form the eight cubes into a single 

 cube in such wise that contiguous faces of the cubes are similarly 

 coloured, and also so that the resulting cube has the appearance 

 of the selected cube in regard to the colouring of its faces. To 

 each cube of the thirty belong in this way eight other cubes, the 

 selection of the eight cubes being unique. For the examination 

 of this property the .selected cube is transformed into an 

 octahedron by j 'ining the middle point of each face to the 

 middle points of the adjacent faces ; a regular octahedron with 

 six differently coloured sum nits is thus obtained. Each tri- 

 angular face is det^-rmined by three differently coloured summits, 

 and exactly eight other octahedra are obtained by circular sub- 

 stitutions performed on the three colours which determine a 

 face ; in regard to the eight faces there are eight clock-wi-eanH 

 eight counter clock-w ise substitutions, but only eight differem 

 octahedra can be obtained. These give the eight cubci 



associated with the selected cube. The eight cubes having been 

 determined, the problem of forming them admits of just two 

 solutions. One solution is — 



Upper Four Cuues. 



The other solution is obtained by interchanging clock-wise 

 and counter clock-wise rotations of octahedral faces. Other 

 miereating properties of these cubes are examined in the 

 paper. 



NO. I217, VOL. 47] 



