85 



then necessary to invent an algebra of non-commutative symbols. 

 This important step was effected by Professor Boole, for certain 

 classes of symbols, in his well-known and beautiful memoir published 

 in the Transactions of this Society for the year 1844 ; and the ob- 

 ject of the paper which I have now the honour to lay before the 

 Society is to perfect and develope the methods there employed. 



For this purpose I have constructed systems of multiplication and 

 division for functions of non-commutative symbols, subject to the 

 same laws of combination as those assumed in Professor Boole's 

 memoir ; and I thus arrive at equations of great utility in the inte- 

 gration of linear differential equations with variable coefficients. 



I then proceed to develope certain general theorems, which will, I 

 hope, be found interesting. I have applied the methods of multi- 

 plication, as just explained, to deduce theorems for non-commutative 

 symbols analogous to the binomial and multinomial theorems of* 

 ordinary algebra. 



Lastly, I have shown how to employ the equations deduced in the 

 earlier part of this paper in the integration of linear differential 

 equations. I have, for this purpose, made use of methods closely 

 resembling the method of divisors which has so long been used in 

 resolving ordinary algebraical equations. The whole paper will, I 

 hope, be found to be a step upwards in the important subject of 

 which it treats. 



I shall just observe, that the symbolical combinations used in this 

 paper may also be applied to the calculus of finite differences, as 

 may be seen in Professor Boole's memoir. 



II. " On the Properties of Liquid Carbonic Acid." By GEORGE 

 GORE, Esq. Communicated by Professor TYNDALL. Re- 

 ceived January 17, 1861. 



(Abstract.) 



In this communication the author has shown how a small quantity 

 of liquid carbonic acid may be readily and safely prepared in glass 

 tubes closed by stoppers of gutta percha, and be brought in a pure 

 state into contact with any solid substance upon which it may be 

 desired to ascertain its chemical or solvent action, or be submitted 



VOL. XI. H 



