On the Summation of a certain Factorial Expression. 419 



zole perfectly definite in its properties, but capable of passing 

 into the latter hydrocarbon by treatment with such powerful 

 reagents as nitric and nitro-sulphuric acids. The existence of at 

 least two distinct yet very similar sulphovinic acids, should not 

 be forgotten ; and, as I have shown* some time ago, these acids 

 have their parallels in the methyle series. The derivation of 

 these isomeric acids can, however, be represented by rational 

 formulae ; while the transformations of turpentine, styrole, and 

 other similar bodies are still obscure. And an observation which 

 I have made throws, I believe, some light upon these metamor- 

 phoses. Cymole prepared from oil of cumin and purified by di- 

 stillation from sodium, boils, according to my determination, at 

 170°' 7; while camphogene, an isomer of cymole obtained from 

 camphor by the action of fused chloride of zinc, boils at about 

 175°. This isomer of cymole has, however, an odour very dif- 

 ferent from that of the natural cymole. But Gerhardt, in his 

 treatise on Organic Chemistry, has the following remark : — " Le 

 cymenc qu'on obtient par la metamorphose du caraphre ne pre- 

 sente pas Fodeur citronnee du cymene naturel ; mais on pent 

 communiquer a ce dernier la meme odeur qu'au cymene factice, 

 en le traitant a chaud par Facide sulphurique concentre, et en 

 Pen separant de nouveau par Paddition de Feau." I have, in 

 fact, found that natural cymole, boiling at 170°' 7, passes, by 

 treatment with sulphuric acid, into an oil undistinguishable in 

 any respect from camphogene, the artificial cymole, and boiling, 

 like it, at 175° or 176°. 



The circumstances in which the remarkable metamorphoses of 

 cymole, &c. are efi"ected, and the peculiarities of the products, 

 seem to point to the possible equivalency of isomerism among 

 compounds to allotropy among elements, in many instances at 

 least. 



Lincoln College, Oxford, 

 May 1857. 



LVII. Note on the Summation of a certain Factorial Eoopression. 

 By A. Cayley, Esq,-\ 



MR. KIRKMAN some months ago communicated to me a 

 formula for the double summation of a factorial expres- 

 sion, to which formula he had been led by his researches on the 

 partition of polygons. The formula in a slightly altered form is 

 as follows : viz. 



* Phil. Mag. July 1855, and January 1856. 

 t Communicated by the Author. 



