84 



were pointed out at the commencement of the paper as forming the 

 essentials of an eye, and considers that it affords a contradiction to the 

 position of Agassiz, "That every great type is formed on a di- 

 stinct plan, so peculiar, indeed, that homologies cannot be extended 

 from one type to the other, but are strictly limited to each of them." 

 On the contrary, the eye of an Asterias is formed on the same plan 

 as that of a Planaria and a Daphnia ; and the eye of the leech pos- 

 sesses the same parts as that of the Helix, while the similarity of 

 the eye of the cephalopod to that of the vertebrata is obvious to all. 



January^, 1861. 



Major-General SABINE, R.A., Treasurer and Vice-President, 

 in the Chair. 



The following communications were read : 



I. "On the Calculus of Symbols, with Applications to the 

 Theory of Differential Equations." By W. H. A. RUSSELL, 

 A.B. Communicated by ARTHUR CAYLEY, Esq. Re- 

 ceived December 20, 1860. 



(Abstract.) 



The calculus of generating functions, discovered by Laplace, 

 was, as is well known, highly instrumental in calling the attention of 

 mathematicians to the analogy which exists between differentials and 

 powers. This analogy was perceived at length to involve an essential 

 identity, and several analysts devoted themselves to the improve- 

 ment of the new methods of calculation which were thus called into 

 existence. For a long time the modes of combination assumed to 

 exist between different classes of symbols were those of ordinary 

 algebra ; and this sufficed for investigations respecting functions of 

 differential coefficients and constants, and consequently for the inte- 

 gration of linear differential equations, with constant coefficients. 

 Thp laws of combination of ordinary algebraical symbols may be 

 divided into the commutative and distributive laws ; and the number 

 of symbols in the higher branches of mathematics, which are com- 

 mutative with respect to one another, is very small. It became 



