T 



72 Royal Society : — 



The more important conclusions to which my experiments have 

 led mc arc these : — 



I. That by the action of water upon sulphate of methyle, 

 /S-sulphomcthylic acid is produced. 



II. That in'the action of water upon sulphomethylate of baryta, 

 sulphate of methyle is formed, and that then the decomposition 

 proceeds according to (I.). 



TIL That sulphomethylate of rethyle yields, by the action of 

 water, methylic and jcthylic alcohols, as well as /3-sulphomethylic 

 and parathionic acids. 

 December 1855. 



IX.' Proceedings of Learned Societies. 



ROYAL SOCIETY. 



[Continued from vol. x. p. 456.] 

 June 21. — The Lord Wrottesley, President, in the Chair. 



HE following communications were read : — 



On the Enumeration of a,'-edra having an (a-— l)-gonal Face, 

 and all their Summits Triedral." By the Rev. Thomas P. Kirk- 

 man, A.M. 



The object of the paper is to enumerate the .i-edra which have an 

 (^— l)-gonal face, and all their summits triedral; or, what is the 

 same thing, to find the number of the a-acra which have an (a.'— 1)- 

 edral summit, and all their faces triangular. 



Every .r-edron having an (a-— l)-gonal face has at least two trian- 

 gular faces. Let A be an ;r-edron having all its summits triedral, 

 and having about its (x — l)-gonal face k triangular faces. Suppose 

 all these triangles to become infinitely small ; there arises an (x—Ic)- 

 edron B, having an (.r — Z:— l)-gonal face, and all its summits tri- 

 edral. B will have k' triangular faces, k' being not le.'-s than two, 

 nor greater than k. And there is no other (.r — A)-edron but B, 

 which can arise from the vanishing of all the k triangles of A ; i. e. 

 there is no (,r — A:)-edron but B, from which A can be cut by re- 

 moving k of the summits of B in such a way as to leave none of its 

 A' triangles untouched. 



If we next suppose the k' triangles of B to vanish, there will arise 

 an (cT — A— A')-edron C, having an (a-— A— A' — l)-gonal face, all its 

 summits triedral, and k" triangular faces, k" not <'2, nor >k'. And 

 thus we shall at last reduce our .z'-edron, either to a tetraedron, or 

 to a pentaedron having triedral summits. 



All .r-edra here considered fall into six varieties, differing in the 

 sequence of the x—l faces that are collateral with the (.r— l)-gonal 

 base. They are either irreversible, as the octaedron 6435443, the 

 seven faces about the base reading differently both backwards and 

 forwards from every face ; or doubly irreversible, as the heptaedron 



