148 Algebra. 



19. Extension op Formula (b) of Chapter II to In- 

 commensurable Indices. 



With the same notation as in the previous article we hav^e 



(a^)y^a'^y (i) 



hence X\vi\ ( a"" y —X\m. a^-'' (2) 



and by Art. 7 lim ^"-^=^'"' (3) 



lyCt x=n-\-ii and as x may be made to differ from its limit ;/ by 

 an amount as small as we please, u may be made as small as we 

 please, or the limit of u is zero. 



Now, by Art. 17, (a '/ = (a"'"/ = (a" a" / (4) 



and because y is commensurable 



(a"a"/=(a"r(a"r = (a"ra"\ (5) 



Substitute in (4) and we get 



(a'-r=(a'^ra"^\ (6) 



hence lim (^^"/ =lim [f^^" /«"-^], Art. 7, (7) 



and lim l(a"ya"y'] = lim (a"/ lim ci"-\ Art. 10 (8) 



therefore lim (^«'^/= lim ('«"/" lim <a;"'. (g) 



But since -y approaches r, therefore 



lim (a")'^'=(a")'\ (10) 



and because // approaches o and _}' approaches r, and hence uj/ 

 approaches o, therefore 



lim a"'"=a"=i. (11) 



Substitute for the right-hand member of (9) the values found in 

 (10) and (11) and we get 



lim (a' y=^(a" )'-. (12) 



Substitute for the two sides of equation (2) the values found in 

 (12) and (3) respectively and we obtain 



fa" )'' =a"''. 



20. Extension of Formula (c) of Chapter II, to In- 

 commensurable Indices. 



With the same notation as in the two previous articles we have 



a"" —a'' =a^''-^' (i) 



Since lim x=n and lim r=r, 



lim (x—j')^=?i—r 

 and lim a"' =a" , lim a\==a'' 



and lim <^'~'=a"~'. 



