42 -^rt- 4.— M. Kuniytnla : 



When /' < .T, X = a is the positive root of tl^e equation 



where {> = xy{x). 



This equation lias, for sufficiently large fixed values of /, one and 

 only one positive root a. Hence tjie function /(re) changes its sign 

 when X passes through the value o.. 



Now, since <y'\d) > 0, yip) > 0, y\d) > 0, a\d) + / = 0, 

 we have 



--^^^^':^^I^^^^ -s. -(. + ,) (0<,<e) 



y\d + B) 

 y'(d) ^ 



^ ^"mr{(f)-^y'(d)] 



y\d) 

 Since y<\, we have 



dy'{ä)^^r,d), 



and, since e<^, -ï'{0<y{d). 



Hence ultimately riß)—^ r'{6) > 0, 



and therefore f{d + e) > 0. 



Thus f{d) and f(ß + ^) have ultimately tJie same sign. 

 Therefore it follows that <: d + e < a. 



Next, when ,■' = Ax{l—J)), r = a is the positive root of tjie 

 equation 



/(.r, = {1 + v{^)Yy^^ - '^'(■^)- / = 0. 



