22 AI.GEBRA. 



fore, and one which has no meaning according to the definition of 

 a power of a number. But a° being i, we know that 



a 

 and so the equation, 



would be true if a~^ were equal to -. 



a 



If we could take one step in this way we ought to be able to 



take two or three or, indeed, any number, and if we could do this 



we could get, in addition to the above, the following equations : 



etc. 

 As we have seen, the first of these equations would be true if 



a~'=~, and from this the secoyid would be true if a~^=~-, and 

 a a 



from this the third would be true if ar^-=~-. etc., and the set of 



a? 



equations just written might be carried just as far as we please if 



a'' 

 q being any whole number. 



Let us examine the effect of this supposition on our formulas 



(a), (b), (c), (d). 



17. If we wish the quotient a"-^-^'' we are directed in Art. 7 

 to use formula (c) if ?^>r, and formula (d) if n<Cr. 

 Suppose «<r and r—7i=^q, then formula (d) gives 



a" 

 But if we should try to use formula (c) in this case we would 

 get 



and this would be true if 



a"' 

 so that, if we could use a quantity with a negative exponent, then 



