Theory of Indices. 25 



Third, suppose both exponents are negative and let 7i=—s and 

 r^=—q, then 



^-'^-'=r.-7 = 



•=<1! 



But by substituting in the formula, 



so that formula (b) may be used when both exponents are neg- 

 ative. 



21. Thus we see that if we interpret a~'^ as being — , a being 



any number whatever (not zero), and q being any whole number, 

 the exponents in all our formulas may be any whole numbers, 

 positive or negative, and this makes our formulas considerably 

 more general than they were before. 



Now, because the supposition «~^=— leads to no inconsistency 



it is permissible, and because it gives greater generality to our 

 formulas it is advantageous. 



Therefore we adopt the equation a~^=- — as defining the mean- 

 ing of a"'^ . 



22. Since a~'- = ~, and therefore -—,=«'', it follows that in 

 a'' a ^ 



any fraction 2iX\y fa dor may be transferred from the numerator to 



the denominator, or vice versa, by simply changing the sign of the 



exponent. 



Hence, if in formula (c) we transfer a" from the denominator to 



a'^ 

 the numerator, — becomes a"a~\ which by formula (a) equals 



a'' 



^"-''; so that formulas (a) and (cj are really identical, but, for the 

 sake of convenience, both are retained. 

 A— .3 



