Theory of Indices. 31 



Third case. 



_ >■ /_ _ '■ _ / _ '■ _ /^. 

 a "-T-a'f=a "a '/=a " '-', 



and by direct substitution in the formula we also get 



a "-r-af=a " ^. 

 Fourth case. 



_ il _ /^ -.It _ .1 4_ Z 

 a '• -^a ^=a '* af =^a " f , 



and by direct substitution in the formula we also get 



_Ji _A _5_ + A ' 

 a "-.-a '^ =a " f. 



Thus we see that by using fractional exponents according to 

 the suggestion before obtained, the result of dividing one frac- 

 tional power of a by another fractional power of a is, in every 

 case, in perfect accord with formula (c). 



31. Now, because the suppositions 



a" ={Vay and a " = • -^ -- 

 (Va) 



lead to no inconsistency they are admissible, and because they 

 give greater generality to our formulas they are advantageous. 

 Therefore we adopt these equations to define the meaning of quan- 

 tities affected with fractional exponents. 



32. The formulas (a), (b), (c) are now so generalized by the 

 above definitions that they can be used when the exponents are 

 any positive or negative whole numbers or fractions, and it might 

 naturally be asked, is this the greatest generality of which they 

 are capable ? 



■ Excluding the so-called imaginaries, there is no kind of alge- 

 braic numbers not yet discussed except incommensurable num- 

 bers, and the consideration of quantities affected with incommen- 

 surable indices is reserv^ed for chapter XI. In the meantime, 

 however, it should be remembered that the formulas are to be 

 used only when the indices are commensurable. 



