Oscillatin<( Dirichlet's Integrals. 55 



Now m = tf^f2%)j - vpi^wr ' 



and hence, observing that ;. = — (T'(d), we have 



y(«) V{^rr-(d)} __ V{2<r"m . 



<P{d) k -o'{d) ^ 



since ^a" < a'. Thus Ave obtain 



<p{a) < <p{d) 



as .^-»x . 



Thus we have completely proved the following proposition. 



Let x<p<X's/(r" or (> =^ Ax[\—{>{x)], ivhere A>0, p>0 cuid 

 |ö < 1. Then 



a{l-a'{a)] ^ d^{1o"{d)] 



as À->cD , a and d being functions of À determined respectively by the 

 equations 



da L a[À — (T\a)\ J 



33- In the above arguments, except in a few special cases, 

 the whole thing depends on the fundamental Lemma 4. We can 

 also prove the same proposition, with exception of a few special 

 cases, by a more direct method without recourse to this lemma. 

 The principal object of the method is to find such asymptotic 

 expressions in terms of À for the functions <p(a} and </>(d), which are 

 of convenient forms for the purpose of comparing their order of 

 magnitude as >^^oo . The analysis is not very difficult and I 

 content myself with giving only the following results. 



Du Bois-Reymond proved^ that, // y be the root of the 

 equation 



* Math. Annalen ßd. VIII (1875), pp. 394 et neq. Du Bois-Eeymond does not state clearly 

 the conditions to which his functions are subjected. 



