Theory of Limits. 147 



than unity a'^^a''^a\ so whether a is greater or less than unit}^ 

 a'^is intermediate between «*and a^ . 



Now consider x and j' variables, but always commensurable, 

 and let x increase and v decrease, and suppose them \o approach 

 the same incommensurable limit ?i. As x andjj/ are commensur- 

 able, ^'and «^have definite meaniiigs, and as x ahd jj/ approach 

 equality, (one increasing and the other decreasing), a' and a-^also 

 approach equality, or in other words there is some quantity be- 

 tween a^'a.nd «^from which each of these quantities may be made 

 to differ by an amount as small as we please. 



But a' and a-'' can never become equal, since x and _y cannot 

 become equal, hence each of these quantities approaches the same 

 limit. 



Since we have now proved that both ^^''and a^ approach a limit, 

 as X and y themselves approach a limit, we may if we choose 

 neglect J and a^and fix our attention upon x and a* remembering, 

 however, that x varies just as it varied before, and hence just as 

 before a"- approaches a limit and indeed the same limit. 



This limit we will represent by a" . Thus we have a meaning 

 for «" where n is incommensurable, viz: it is the limit approached 

 by a^ (x being commensurable) as x approaches a limit w. 



18. Extension of Formula (a) of Chapter II to Incom- 

 mensurable Indices. 



So long as x and y are commensurable we know that 



lyCt X approach an incommensurable limit ;/, and y approach an 

 incommensurable limit r. 



Then x-\-y approaches a limit n-\-r which is usually incom- 

 mensurable, but may possibly be commensurable. Also a"- ap- 

 proaches a limit a" , a' approaches a limit a'' and a' ^ approaches a 

 limit a"+". 



Now in equation (i) the left-hand member is one variable, and 

 the right-hand member is another which is equal to the first. 

 Hence, Art. 7, lim a^a^^Xww «^+''. 



But the left-hand member is the product of two variables, hence 

 by Art. 10, lim (a"" a' )=\\ixi a' lim of" or lim (a' a^ )=a" a'^ , hence 

 a" a' =a"'^''. 



