198 THIRD REPORT — 18S3. 



have been suggested only by the corresponding rules in arith- 

 metical algebra. They cannot be said to he founded upon them, 

 for they are not deducible from them ; for though the opera- 

 tions of addition and subtraction, in their arithmetical sense, 

 are applicable to all quantities of the same kind, yet they ne- 

 cessarily require a different meaning when applied to quanti- 

 ties which are different in their nature, whether that difference 

 consists in the kind of quantity expressed by the unaffected 

 symbols, or in the different signs of affection of symbols de- 

 noting the same quantity ; neither does it necessarily follow 

 that in such cases there exists any interpretation which can be 

 given of the operations, which is competent to satisfy the re- 

 quired symbolical conditions. It is for such reasons that the 

 investigation of such interpretations, when they are discover- 

 able, becomes one of the most important and most essential of 

 the deductive processes which are required in algebra and its 

 applications. 



Supposing that all the operations which are required to be 

 performed in algebra are capable of being symbolically de- 

 noted, the results of those operations will constitute what are 

 called equivalent forms, the discovery and determination of 

 which form the principal business of algebra. The greatest 

 part of such equivalent forms result from the direct applica- 

 tion of the rules for the fundamental operations of algebra, 

 when these rules regard symbolical combinations only : but 

 in other cases, the operations which produce them being nei- 

 ther previously defined nor reduced to symbolical rules, unless 

 for some specific values of the symbols, we are compelled to 

 resort, as we have already done in the discovery and assump- 

 tion of the fundamental rules of algebra themselves, to the re- 

 sults obtained for such specific values, for the purpose of dis- 

 covering the rules which detei-mine the symbolical nature of 

 the operation for all values of the symbols. As this principle, 

 which may be termed the princijile of the permanence of equi- 

 valent forms, constitutes the real foundation of all the rules of 

 symbolical algebra, when viewed in connexion with arithmeti- 

 cal algebra considered as a science of suggestion, it may be 

 proper to express it in its most general form, so that its autho- 

 rity may be distinctly appealed to, and some of the most im- 

 portant of its consequences may be pointed out. 



Direct proposition : 



Whatever form is algebraically equivalent to another ivhen 

 expressed in general symbols, must continue to be equivalent, 

 whatever those symbols denote. 



Converse proposition : 



