REPORT ON CERTAIN BRANCHES OF ANALYSIS. 201 



applications, such additional signs * are assumed, and of such a 

 symbolical form, as those applications may render necessary. 

 We call those rules, or their equivalent symbolical consequences, 

 assumptions, in as much as they are not deducible as conclusions 

 from any previous knowledge of those operations which have 

 corresponding names : and we might call them arbitrary as- 

 sumptions, in as much as they are arbitrarily imposed upon a 

 science of symbols and their combinations, which might be 

 adapted to any other assumed system of consistent rules. In 

 the assumption, therefore, of a system of rules such as will make 

 its symbolical conclusions necessarily coincident with those of 

 arithmetical algebra, as far as they can exist in common, we in 

 no respect derogate from the authority or completeness of sym- 

 bolical algebra, considered with reference to its own conclu- 

 sions and to their connexion with each other, at the same time 

 that we give to them a meaning and an application which they 

 would not otherwise possess. 



It follows from this view of the relation of arithmetical and 

 symbolical algebra, that all the results of arithmetical algebra 

 which are general in form are true likewise in symbolical 

 algebra, whatever the symbols may denote. This conclusion 

 may be said to be true in virtue of the principle of the perma- 

 nence of equivalent forms, or rather it may be said to be the 

 proper expression of that principle. Its consequences are most 

 important, as far as the investigation of the fundamental pro- 

 positions of the science are concerned, in as much as it enables 

 us to investigate them in the most simple cases, when the 

 operations which produce them are perfectly defined and un- 

 derstood, and when the general symbols denote positive whole 

 numbers. If the conclusions thus obtained do not involve in 

 their expression any condition which is essentially connected 

 with the specific values of the symbols, they may be at once 

 transferred to symbolical algebra, and considered as true for 

 all values of the symbols whatsoever^. 



Thus, coeflicients in arithmetical algebra, such as m in m a, 

 which are general in form, lead to the interpretation of such 



* There is no necessary limit to the multiplication of such signs : the signs 

 + . — ) (1)" ^"d (—1)" and their equivalents (for the symbolical form of such 

 signs is not arbitrary), comprehend all those signs of affection which are re- 

 quired by those operations with which we are at present acquainted. 



t Some formulEe are essentially arithmetical : of this kind is 1 . 2 . 3 . . . r, 



in which r must be a whole number. The formula "^ ("» — 1) •••("'- ^ + 1) 



1 . 2 . . . ;• 

 is symbolical with respect to m, but arithmetical with respect to r. Such cases, 

 and their extension to general values of r, will be more particularly considered 

 hereafter. 



