34 -^I't. 4.— M. Kuniyeda : 



Hence 

 (66) 



T(.. N _ 1 /' .a?-r* COS (a+r/?) g[Q;-i;ç + «ri"''siii(a + î?)]/. # 



^^''^~ 2:^^.1 _ ^._,) ' (1 - rxe-^0"^' 



where e denotes the difference of the angle OAB and a right- 

 angle, namely 



~-e= zlOAB = ^OAB, 



and we ohserve that lim e = 0. 



ri->0 



Now, if cos(« + g^)>0 in any part of the range 



then e""''^ " cos {<x+ q<?) tends exponentially to infinity as n-^O, so that 

 I(ri) does not necessarily tend to zero. Hence we shall put aside 

 this case and confine ourselves to the case in which 



cos (« + (7^) s 0, 



or 



(67) (4m+l)^ s« + g^s fém + S)^ (-!- + £ s ^ s |._e), 



m being a positive integer or zero. 



If we observe that, by hypothesis, 



s a < 2;r, 



and that the condition (07) is to be satisfied by all values of f in 

 the interval — |-+e s ^ s |.~e, where e takes any value corres- 

 ponding to vi which tends to zero, it can easdy bo inferred that 



i < (7 s 1, 



I a + g)-^-^«^(3-.i)|-. 



Now we can prove the lennna. 



Lemma 11. If o < q m l, (l+q)^ ^ a ^ (3-^) ^ and 



p < 1 + q, then 



lim I{r,) = 0. 



