REPORT ON CERTAIN BRANCHES OF ANALYSIS. 199 



Whatever equivalent form is discoverable in arithmetical 

 algebra considered as the science of suggestion, when the sym- 

 bols are general in their form, though specif c in their value, 

 will continue to be an equivalent form when the symbols are 

 general in their nature as well as in their form *. 



The direct proposition must be true, since the laws of com- 

 bination of symbols by which such equivalent forms are de- 

 duced, have no reference to the specific values of the symbols. 



The converse proposition must be true, for the following 

 reasons : 



If there be an equivalent form when the symbols are general 

 in their nature as well as in their form, it must coincide with 

 the form discovered and proved in arithmetical algebra, where 

 the symbols are general in their form but specific in their na- 

 ture ; for in passing from the first to the second, no change in 

 its form can take place by the first proposition. 



Secondly, we may assume the existence of such an equivalent 

 form in symbols which are general both in their form and in 

 their nature, since it will satisfy the only condition to which 

 all such forms are subject, which is, that of perfect coincidence 

 with the results of arithmetical algebra, as far as such results 

 are common to both sciences. 



Equivalent forms may be said to have a necessary existence 

 when the operation which produces them admits of being de- 

 fined, or the rules for performing it of being expressly laid 

 down : in all other cases their existence may be said to be 

 conventional or assumed. Such conventional results, however, 

 are as much real results as those which have a necessary ex- 

 istence, in as much as they satisfy the only condition of their 

 existence, which the principle of the permanence of equivalent 

 forms imposes upon them : thus, the series for (1 + ;r)" has a 

 necessary existence whenever the nature of the operation upon 

 \ + X which it indicates can be defined ; that is, when « is a 

 whole or a fractional, a positive or negative, number f ; but for 

 all other values of n, where no previous definition or interpre- 

 tation of the nature of the operation which connects it with its 

 equivalent series can be given, then its existence is conventional 

 only, though, symbolically speaking, it is equally entitled to be 

 considered as an equivalent form in one case as in the other. 



It is evident that a system of symbolical algebra might be 



• Peacock's Algebra, Art. 132. 



+ The meaning of (1 + «)" cannot properly be said to be defined when n 

 is a fractional number, whether positive or negative, or a negative whole num- 

 ber, but to be ascertained by interpretation conformably to the principle of 

 the permanence of equivalent forms. 



