Theory of Indices. 21 



J3. Third, a"^a'-=a''-''. (c) 



If we make n=r we get 



a"^a"=a\ 

 and this is true if a°= 1 . 



Again, if we make r=o we get 



and this also is true if <2°= i . 



(d) 



15. Now, because the assumption ^°=i leads to no incon- 

 sistency it is permissible, and because it gives greater generality 

 to our formulas it is advayitageous. 



Therefore we adopt the equation «°= i as defining the meaning 

 oia\ 



16. The question naturally arises, is there any way whereby 

 we may give still greater generality to our formulas ? 



Let us look again at our process of successive division. 

 We have already obtained the equations, 



a^-r-a=^d', 

 a'-^a=-a, 



fTow, if, by the same rule, (viz : subtract one each time from 

 the exponent,) we attempt to take another step we get 



a''-^a=a~\ 

 Here, again, we have a symbol a~^ that has not been used be- 



