6 DIFFERENTIAL AND INTEGRAL CALCULUS. 



1, whatever may be the value of n ; hence since li has 



here to diminish indefinitely , - must be numerically less 







than 1, before arriving at the ultimate hypothesis A = 0, or 



-II 



therefore -~ - = nx n ~ l \ 1 + ^ h - 



Li *' 



The quantity in brackets terminates when n is an integer, 

 but has an infinite number of terms when n is fractional 



or negative, but is in alL cases convergent when - < 1, 



and reduces to 1 when h and therefore -=0. hence 



x 



dy 



~ = nx \ 

 ax 



or -- = a x x limit of 



dx 



Now a h = 1 + h loga + ^- (log a) 2 +... ; 



L^ 



a* - 1 f 7t , /** 1 



therefore = log a l-f- lo g + (loga) 2 -f ... . 



But the series 



ti , h 



h . (h . \ fit . v 



is >1 and <1 + - loga-f I- logal -h I - logaj 4..., 



and therefore > 1 and < , , 



I--loga 



when 7^ has been so far diminished that - loga<l, and 



