60 A-rt. 4. — M. Kuniyeda : 



lU) = I t/^x) cos fr(x) ^—âx, 



Jo X 



I(^X) = I J>{x) cos(t[x) — - dx. 



The?>e integrals are convergent, if f> < n'. 



Put K^) = 44 • 



Then t{x) is ultimately monotonie and tends to zero as x^O ; and 

 we assume that ç is chosen sufficiently small to ensure that e(a;) is 

 monotonie in the interval < a- < c. 



We may write 



^(''0 = / K^) K^) *2C>s (t(x) dx 



= I ^(x)f(x) sin ÀX dx, 



./ 



where 



j,.^ ^ p(x) cos ajx) ^ 



SO that f(/) = / /(«) sin ?.x dx. 



./ 



Now, corresponding to any prescrihed positive number d, however 

 small, there always exists a positive number ?', independent of /, 

 such that 



< £(ç') < O, (0 < ç' < ç). 



We have 



/(;.) = (^ r + / '\ e(x'f(x) sin Xx dx 



say. Then, in the integral 



J"(2)(A) =y ' e{x)/{x) sin Ax dx, 



