Royal Society. 459 



St. Sampson's quay were so shaken, that glass vessels situated on 

 various parts were thrown off. Two gentlemen engaged in Daguer- 

 reotype experiments on the ramparts of a fortification founded on a 

 solid granite rock, felt the whole to vibrate. The crews of sailing- 

 vessels beating up in the " roads," also felt the shock ; those below 

 rushing on deck under the impression that the vessels had struck on 

 a rock. 



The testimony of a great number of witnesses leaves no doubt as 

 to the distinctness and strength of the shock. It was also felt, though 

 in A slighter degree, in the neighbourhood of St. Malo, and near 

 Brixham in Devonshire. 



January 18.—" On a new Method of Analysis." By George Boole, 

 Esq. Communicated by S. Hunter Christie, Esq., Sec. R.S., &c. 



The purport of this paper is to exhibit a new form of analysis, 

 and to found upon it a new theory of Linear Differential Equations^ 

 and of Generating Functions. The peculiarity in the form of the 

 analysis consists in the linear differential equation, instead of being 

 represented, as it has hitherto been, under the type 



■ft sf'~ l 



x fl JLiL + Xl d *■ +X n ti = X, 



dx n x dx n ~ l n 



X , X„ &c. being functions of the indepehdent variable x, being 

 exhibited in the form 



/ (D)«+/ I (D)^i» + /.(D)«*i« = U; 



in which e = x, and/ (D), f x (D), &c. imply functional combina- 

 tions of the symbol D, which, for the sake of simplicity, is written in 



place of — . This the author calls the exponential form of the 



equation J and he, in like manner, designates the analogous forms 

 of partial and of simultaneous equations. What he conceives to be 

 the great and peculiar advantage of the exponential form, both as 

 respects the solution of linear differential equations, and the theory 

 of generating functions, is that the necessary developments, trans- 

 formations and reductions are immediately effected by theorems the 

 expression of which is independent of the forms of the functions 

 f Q (D),/i (D), <&c. Accordingly it may be shown that various for- 

 mulas which have been given for the solution of linear differential 

 equations, with those in which Laplace's theory of generating func- 

 tions is comprised, interpreted into the language of the author, are 

 but special cases of theorems dependent on the exponential form 

 above stated, and Which are susceptible of universal application. 



The common method of effecting the integration of linear differ- 

 ential equations in series fails when the equation determining the 

 lowest index of the development has equal or imaginary roots. In 

 a particular class of such equations of the second order, Euler has 

 sho\vn that log. x is involved in the expression of the complete inte- 

 gral : but this appears to be merely a successful assumption ; artd 

 the rule of integration demonstrated in the present paper admits of 

 tto such cases of exception whatever. 



The finite solution of linear differential equations may be attempted 



