2Q Art. 4.-M. Knniyeda : 



important properties concerning the variations of the functions 



P(x) p{x) 



x[X-a'{x)} ' œ{). + a'{x)} 



for .-^ufticiently large values of /, provided that 



<r > l{^/x), 

 and X < fj < xff'.^ 



I will give two more lemmas of a similar nature, concerning 

 the variations of these two functions, for the case in Avhich 



<y > l{\lx), 

 and x' <p<x. 



16. Lemma 2. Let ^ > /(i/a;). 



(i) If f><x, or if f^ = Ax[\+p[x)], 

 where A is a positive constant and 



the7i the function 



x{Ä—o') 



is a steadily increasing function of x tJirouijhout the interval 0<ic<ç. 



00 // i> = Ax{\-Jix)], 



where A>Q, p>0 and o<\, then the function <f has, for sufficiently 

 large fixed values of /, one and only one stationary value in the range 

 0<a;<ç, ivhich is a, maxiimiyn and tends to zero as x^oo . 



Proof. In the case (i), -^ is evidently a steadily increasing 

 function of x (or a constant when /v^O) in the interval (0» f) and 

 so also is the function 



* In the followint'- investigation of Case (C), we shall assume that 



P >> 0, a > 0. 



Wc can easily see that, liy this assuiiiptiou, no loss «if i^euerality will l)o introduced. 



