1865.] Mr. W. H. L. Russell on Symbolical Expansions. 329 



The chief specimens described in the memoir are eight in number, and 

 were found in the lower divisions of the Lancashire and Yorkshire coal- 

 measures imbedded in calcareous nodules occurring in seams of coal. 



No. 1, Diploxylon cycadoldeum, was from the first-named district, and 

 the same locality as the Trigonocarpon, described by Dr. J. D. Hooker, 

 F.R.S., and the author, in a memoir on the structure of certain limestone 

 nodules inclosed in seams of bituminous coal, with a description of some 

 Trigonocarpons contained therein*, and the other seven (Sigillaria vascu- 

 laris) were from the same seam of coal in the lower coal-measures in which 

 the specimens described in a paper entitled " On some Fossil Plants showing 

 structure from the Lower Coal-measures of Lancashire " f, were met with, 

 but from a different locality in Yorkshire. 



III. " On Symbolical Expansions." By W. H. L. RUSSELL, Esq., 

 A.B. Communicated by Prof. STOKES, Sec. R.S. Received 

 May 13, 1865. 



Among the papers on symbolical algebra by the lamented Professor 

 Boole, there is one on the Theory of Development, published in the fourth 

 volume of the ' Cambridge Mathematical Journal.' The expansion of 



f ( x +-r- ) is there given in a very elegant form. I am desirous to ter- 



minate my own investigations on the Calculus of Symbols by pointing out 

 the connexion of the binomial theorems given in my first paper on this sub- 

 ject with the expansions due to Professor Boole, and propose with that view 



to expand / ( x+x^-\ in terms of , which will be sufficient to indicate 

 the general method. When the term of the expansion which does not 

 contain is known, the other terms are easily found by a method given 



by Professor Boole in the paper I have just mentioned. The main object 

 of the present paper, therefore, will be to ascertain that part of the expan- 



sion of f (x-\-x ) which does not contain . 

 \ dx) ax 



Putting, as usual, p for (#) and TT for x , the expression becomes 



f (p +TT). Our first object must be to ascertain that part of the expansion 

 of (P+TT)" which is independent of (TT), from whence we may easily deduce 

 the corresponding portion of / (P + TT). Now by a former paper the part 

 of (P + TT)", independent of TT, will be 



+ &c. +S(w 



* Philosophical Transactions, 1855, p. 149. 



t Quarterly Journal of the Geological Society of London for May 1862. 



