210 Royal Society : — 



exceeds by unity the sum of the number of edges and of the 

 number of pol3'hedra." 



For P=l, we have Euler's equation S + H = A + 2; and for 

 P = 0, we have a theorem relating to the partition of a polygon; 

 viz. if the polygon is divided into H polygons, and if S be the 

 number of vertices, and A the number of sides, then S + H = A + 1 ; 

 from which it is easy to pass to Euler's equatiou, S + H = A + 2, 

 for polyhedra. I remark that, in the equation S + H = A + 1, 

 H should, in analogy with Cauchy's notation for polyhedra, be 

 replaced by P ; so that we have for a single polygon, 



A = S; 

 and for the partitions of a polygon, 



A = S + P-1: 

 corresponding respectively to Euler's theorem for a single poly- 

 hedron, viz. 



S + H = A + 2; 



and to Cauchy's theorem for the partitions of a polyhedron, viz. 



S + H=A + 2 + (P-l). 



Cauchy's second memoir (pp. 87-98) contains a very beautiful 

 demonstration of the theorem implied in the ninth definition of 

 the eleventh book of Euclid, viz. that two convex polyhedra are 

 equal when they are bounded by the same number of faces equal 

 each to each. 



2 Stone Buil(linp;s, W.C., 

 February 1, 1859. 



XXXIV. Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from p. 147.] 



June 1 7, 1858. — The Lord Wrottesley, President, in the Chair. 



THE following communications were read : — 

 " Researches on the Action of Ammonia on Glyoxal." By Dr. 

 H. Debus. 



If alcohol be slowly oxidized by nitric acid at ordinary tempera- 

 tures, besides other substances, glyoxal, C, H, O,, and glyoxylic acid, 

 Ca H^ O^*, are formed. 



I have continued the investigation of these substances, and beg 

 to lav before the Royal Society some of the more interesting results. 



Glyoxal, of the consistency of ordinary syrup, is mixed with about 

 three times its bulk of strong ammonia, and the mixture kept for 

 twenty minutes at a temperature from 60° to 80° C. The liquid 



* C=12, H = l, = 16.— PhU. Mag. Nov. 1856, tnd Jan. 1857. 



