Dr. Herapath on the Cinchona Alkaloids. . 55 



The object of the memoir is to enumerate the number of such par- 

 titions that can be made with K diapeds in the vertex and k diagonals 

 in the bases of the pj^ramid. By the drawing of k diagonals, the 

 pyramid becomes a (?•+ l)-acral {r + k+ l)-edron, which by the in- 

 troduction of K diapeds becomes a (r-)-K+ l)-acral (?- + /:-|-l)-edron. 

 Such a figure is termed an r-gonous (?--f-K.+ l)-acral {r-\-k+ l)-edron 

 of the first class. The definition of an ?'-gonous a;-edron of the first 

 class is that it contains a discrete r-gony, i. e. K diapeds and k 

 diagonals of which no diaped meets a diagonal, and such that the 

 couvanescence of the K diapeds will form an ?'-ace, and the eva- 

 nescence of the diagonals forms an r-gon. 



If the summits upon the k diagonals be, one or more of them 

 partitioned by K' diapeds, or the faces about the K diapeds be par- 

 titioned by k' diagonals, there arises a mixed r-gony, in which are one 

 or more angles made by a diaped and a diagonal. If such a figure 

 has not a discrete r-gony as well as that mixed one, and has no 

 (?--f-?"')-gony, by the vanesceuce of which the (r-l-r')-pyramid can 

 be obtained, it is an r-gonous a'-edron of the second class. And 

 r-gonous a;-edra of the third class can be obtained by partitioning 

 the faces about the K' diapeds and the summits upon the k' diagonals, 

 in such a manner that no (r-|-r')-gony shall be introduced ; and so 

 on for higher classes of ?'-gonous a?-edra. 



It is proved that every partition proper of the r-pyramid, that is, 

 any (1 4- K) -partitioned r-ace laid on a (1 -)-A-)-partitioned r-gon, is 

 an /'-gouous {r + k-\- l)-acral (r-\-k-\- l)-edron. The number of the 

 (1 -f A-) -partitions of the ?'-gon, and of the (1 +K) -partitions of the 

 r-ace is known by the formiilse given in the author's memoir " On 

 the partitions of the r-gon and r-ace," in the Philosophical Trans- 

 actions, 1857. The present memoir gives the formulae whereby the 

 partitions of the pyramid are determined in terms of those of the 

 /•-gon and r-ace. 



Thus the entire first class of r-gonous >r-edra is enumerated, with- 

 out descending to any classification of polyedra according to the 

 rank of their faces and summits. The enumeration of the second 

 and higher classes will require such classification, which will in- 

 troduce so vast a complexity as to render the further prosecution of 

 the theory of the polyedra, in the opinion of the author, practically 

 impossible by any method deserving the name of scientific generality. 



" Researches on the Cinchona Alkaloids." By W. Bird Hera- 

 path, M.D. Lond., F.R.S.E. 



In the first part of this paper, the author examines the existing 

 tests for discriminating between the various cinchona alkaloids, and 

 points out their insufficiency. In the present part, he shows that 

 the optical characteristics of the iodo-sulphates of the alkaloids 

 quinine and (juinidine arc sufficiently well marked to render the ex- 

 istence of either one of these alkaloids certain, and that although 

 the iodo-sulphate of cinchonidinc is very closely related optically and 

 chemically to the homologous salt of quinine, yet there are sufficient 

 points of dissimilarity to enable us to diagnose between the two ; 

 and, moBCOver, that the jjroduction of this salt is a beautiful means 



