23 Art. 4. -M. Knniycda : 



whence it follows that, for sufficiently large fixed values of >'., the 

 equation (25) lias one, and only one, root. Thus the function <p 

 has one stationary value; and as <p is positive and <p<^, this value 

 is plainly a maximum. 



If the root of the equation (25) is x=a, then the value of f(«) 

 is given by 



^^'^ ^^"^- "^>) a:wxär" 



and, by (25), ;,-.'(«) = ^i^. 



Hence ^<^"^ = ^^'- 



and .,/(.) =-_^(^) = ^:-^, 



which proves the equation (27). 



Since — 7->i> we see that f.«) tends to zero as /^oo 



r 



'J'he ])roof of the lemma is thus completed. 

 17. Lemma 3. Let rr>i{llx). 



( i ) //■ r<x or if f> = Ax[^ +7'(x)], 

 where A is a positive constant and 



then the function 



^ x{X + a') 



