g2 Art. 4. — AI. Kuniyeda : 



for all sufficiently large values of I, K being a constant independ- 

 ent of z'.. For, if not, corresponding to any given A", there would 

 exist some large values of )^ for which 



\m\>K\i{x)\, 



and 



I j-(i)(/i) 1 = I e(ç')[y(/)-/(/)] I > <r) I (z- 1) I i{K) i - 1 0(1) |. 



Hence \J^'\X)\^\k\I{XY - o[\)\,* 



and we obtain 



|j(;OI> K\m\. 



Therefore it follows that 



\s{r)\> K\s{X)\, 



contrar}'^ to our assumption (52). 

 Thus w^e have 



|/(A)i<ZlZ(/)t 



for all sufficiently large values of )^. Hence we have 



|J"<^>(/Î)| < oZ|7(/)i +0(1), 

 and 



\î{k)\<dK\i{?y\+o{\), 



whence it follows that 



l^(;.)l < dK\s{k)\ + 0(1). 



As will l)e seen presently, f I^{â) does not tend to zero as^'^^x. 

 Therefore 





oK + o(l) <: {K+\]o 



by choosing / sufficiently large. Now Jv is a constant independ- 

 ent of ?. and o may be chosen as small as we please. Hence 



* Here K is written for z{V) (K—l) and, as £(^') is a constant independent of )., K in this 

 expression may take any large val vie as we please, 

 t See the next paragraph 37. 



