228 H. m'^coll on the growth and 



secondary importance ; almost any arbitrary symbol of 

 easy formation would have done just as well; but tbis is 

 an exception to tbe general rule. Consider tlie symbol a**, 

 wMcb has been invented as an abbreviation for tbe pro- 

 duct of n equal factors^ eacb equal to a ; that is^ a^ for aa, 

 aP for aaa, and so on. If tbe first of these products, 

 namely aa, were tbe only one that had a tendency to 

 recur, we may be quite sure that mathematicians would 

 remain satisfied with it in its original form, and would 

 never have accepted the innovation a^ as its equivalent. 

 But since aaa, aaaa, &c. have also the same tendency of 

 frequent recurrence, the appropriateness of the symbol 

 selected is evident : the numerical index reminds us of the 

 number of equal factors ; and we are at once provided 

 with a more effective notation for considering the pro- 

 perLies and relations of all expressions that are products of 

 equal factors, as, for instance, in the binomial theorem. 



Let us now examine the raison d'etre of that remark- 

 able class of symbols which were invented at a more 

 advanced stage of the science (by whom I know not), and 

 which give such a wonderful sweep and power to sym- 

 bolical language generally, logical as well as mathematical ; 

 I refer to that class of symbols of which f{x) may be taken 

 as a specimen. This symbol denotes any complex ewpres- 

 sion whatever (mathematical or logical) that contains the 

 simpler expression x, in any relation whatever, as one of its 

 constituents. What was the special need which this symbol 

 was invented to supply ? 



We have often to consider what an expression would 

 become if one of its constituents were taken away and a 

 fresh constituent put into its place, just as people some- 

 times speculate as to what would be the effect upon a 

 ministerial policy if a certain member of the cabinet were 

 to resign and a certain other person appointed in his place. 



