126 The Rev. J. Bayma on Molecular Mechanics. [Feb. 11, 



the breaking-weight upon bridges of this description, as, according to the 

 Jast experiment, the beam broke with 313,000 changes ; or a period of eight 

 years, at the same rate as before, would be sufficient to break it. It is more 

 than probable that the beam had been injured by the previous 3,000,000 

 changes to which it had been subjected ; and assuming this to be true, it 

 would follow that the beam was undergoing a gradual deterioration which 

 must some time, however remote, have terminated in fracture. 



February 11, 1864. 

 Major-General SABINE, President, in the Chair. 

 The following communications were read : — 



I. " On the Calculus of Symbols. — Fourth Memoir. With Applica- 



tions to the Theory of Non-linear Differential Equations. By 

 W. H. L. Russell, A.B. Communicated by Professor Cayley. 

 Received July 31, 1863. 



(Abstract.) 



In the preceding memoirs on the Calculus of Symbols, systems have been 

 constructed for the multiplication and division of non-commutative symbols 

 subject to certain laws of combination ; and these systems suffice for linear 

 differential equations. But when we enter upon the consideration of non- 

 linear equations, we see at once that these methods do not apply. It 

 becomes necessary to invent some fresh mode of calculation, and a new no- 

 tation, in order to bring non-linear functions into a condition which admits 

 of treatment by symbolical algebra. This is the object of the following 

 memoir. Professor Boole has given, in his * Treatise on Differential Equa- 

 tions,' a method due to M. Sarrus, by which we ascertain whether a given 

 non-linear function is a complete differential. This method, as will be seen 

 by anyone who will refer to Professor Boole's treatise, is equivalent to find- 

 ing the conditions that a non-linear function may be externally divisible by 

 the symbol of differentiation. In the following paper I have given a nota- 

 tion by which I obtain the actual expressions for those conditions, and for 

 the symbolical remainders arising in the course of the division, and have 

 extended my investigations to ascertaining the results of the symbolical 

 division of non-linear functions by linear functions of the symbol of differ- 

 entiation. 



II. " On Molecular Mechanics." By the Rev. Joseph Bayma, of 



Stonyhurst College, Lancashire. Communicated by Dr. Sharpey, 

 Sec. R.S. Received January 5, 1864. 



The following pages contain a short account of some speculations on 

 molecular mechanics. They will show how far my plan of molecular 



