557 



which I had the honour to lay before the Society in December 1860, 

 and which has since been published in the ' Philosophical Trans- 

 actions.' I commence this paper with some extensions of the method 

 given in the former memoir for resolving functions of non-commu- 

 tative symbols into binomial factors. I then explain a method, 

 analogous to the process for extracting the square root in ordinary 

 algebra, for resolving such functions into equal factors. I next in- 

 vestigate a process for finding the highest common internal divisor 

 of two functions of non- commutative symbols, or, in other words, of 

 finding if two linear differential equations admit of a common solution. 

 After this, I give a rule for multiplying linear factors of non-commu- 

 tative symbols, analogous to the ordinary algebraical rule for linear 

 algebraical factors. I then resume the consideration of the binomial 

 theorem explained in the former memoir. Two new forms of this 

 binomial theorem are here given ; and the method by which these forms 

 are proved identical will, I hope, be considered an interesting portion 

 of symbolical algebra, and as exhibiting in a remarkable manner its 

 peculiar nature. 



II. " On Internal and External Division in the Calculus of 

 Symbols." By WILLIAM SPOTTISWOODE, Esq., M.A., 

 F.R.S. Received January 8, 1862. 

 (Abstract). 



Continuing my researches in the calculus of symbols, I have been 

 led to investigate the most general case of division, viz. that wherein 

 a function of any degree n in TT is divided, (1) internally, (2) ex- 

 ternally, by another function of any other degree m in TT. The 

 investigations here subjoined give (I) the various terms of the quo- 

 tient, together with their laws of derivation both by actual division 

 and otherwise ; (2) the final remainder, and thence the conditions 

 that the divisor may be a factor, internal or external as the case may 

 be, of the dividend. An example has been added in each case by 

 way of illustrating the processes. A remarkable reciprocal relation 

 subsisting between the functions ($), of the coefficients (</>) of the 

 dividend, and the corresponding functions (^) of the coefficients (^) of 

 the divisor is exhibited, at the end of the paper. 



