Theory of Limits. 145 



First. When n is a positive integer. If in Art. 1 1 we let j/, 

 2, etc., each equal x then b, c, etc., will each equal a, and hence 



\\V[i(xxx . . . )==aaa . . . 

 or lim x"=a". 



Second. When ;/ is a positive fraction, say - .. 



1 

 Let ^-'/=j/, ^ (ij 



then ^'=y^, . (2) 



hence b}^ Art. 7 a—b\ (7,) 



where b is the limit of r; hence 



hence by Art. 7 \\m x'^ —Mm y^—b^. (6) 



But from (^) b^^aJ , 



hence lim jf '''=<2^; 



therefore the theorem is true for any positive exponent whether 



integral or fractional. 



Third. Let n be a negative quantity either integral or frac- 

 tional, say 7i=—s, then x~'=-~; therefore 



X' 



lim x~'=-~=a~\ 

 a' 



hence the theorem is true for any commensurable exponents. 



INCOMMEN.SURABLE POWERS. 



15, In Chapter II we have found that whatever commensur- 

 able numbers are represented by n and r then 



a" .a'^ =a"^'' (a) 



(a'^y^a'"- (b) 



a" -—a'' =a"~'' (c) 



but no meaning has yet been given to quantities with incommen- 

 surable indices. 



The quantity raised to a power is called the Base. In Chapter 

 II, the base was either positive or negative, but the present dis- 

 cussion is confined to the case where the base is positive. 



