( 435 ) 



al which tp" becomes imaginary or the value of x at which 



becomes = 0. We can decide this al once by substituting^ = 



d*A 

 the limiting value for Up'") real, in , . We Imd then: 



(i.i" 



d l A 

 dx* 



d?A t o 3 \a x a a 



dx* a' ( o c 



or 



" n 



{l-xf j V 



c k a, «, 



= -2- v 2 -v(i-^) s -^ 



a? =—? v a-*) J 7 -a-*>, 



a 8 ^ 2 -^ 



As — >'l, and afortiori>(l- a?^, we find — still negative. 



Finally by availing ourselves of the values obtained we shall be 

 able to verity that even if the function («//") is not real over the 

 full extent from x = to ,r. = 'l, and so if our conclusion thai this 

 function must possess a minimum value which is negative, can no 

 longer be considered as proved, there is even in this case also a 



r0()t f 01 . l A ( LJ == o at a value of x which is smaller than ,/;,„ and 

 dx 



which has therefore the former meaning for (<p'"). 

 , d(<p'") 



For the root of 



dx 



a„ — u' a 



c 



holds the equation : 



- = (1— .*•)'" — n ' 2 - i; " (see page 431 



and so if — = x g * is put, the following equation holds: 

 c 



a -l - (\-w g y = (l-x)* - (l-w 7 ) 2 + n* « 



or 



*-! - (1_ % )' = (x g -x) (2 + (n--l) (■>;, I •'•)! 



c 



an( j ;l s ' <2 — (1- -x g )* is positive, (.% — j?) must also be positive, or the 



c 



it '"\ 

 rooi of— - = lies at smaller value of x than thai of the. final 



dx 



