Y4 Art. 4. — M. Kuuiyeda : 



as /^x , wlien X's/n"lrf' •< (> < xa'. 



In the integral 



/(/) = I'' 7>(a-) cos o{x) ^^^^ dx (0 < Çi < ç), 



/ 3? 



we ol)serve tliat ci varies with I; still, if Ave examine the proof of 

 Theorem VI. we can see without difficulty that 



j'{À)<Kj{X). 



Hence 



j(;0 = é(ç)iy(/) -/(;.)] </iW, 



if x^o"la' < /> < x(t'. Therefore we have: 



If (> < xV^"/a', then ^(/) = 0(1//) ; 



if x^r7"/(r' < ;, < xa', then /(/) -^ I^{?^). 



The same argument applies to the otlier integrals. 

 Tluis the lemma is completely proved. 



VIII Km.mpks of Case (6') 

 41- Let us consider the case in which 



-a '>n 



wJiere m is positive, so that 



ni \ cos Xx 



/o X ./o a 



<ia;, 



cZaj, 



dx. 



m 



Since 't = — -^ IQ/x), Theorems XJ and VII are applicahle. 

 Now 



m It 2m 



'a;"" 



