( 197 ) 



/rfaV 2 d*a ,. . . 

 holds; then the point where -- = - a — - lies in the region where 

 1 \dxj o dar 



a 

 the value of - again increases. 

 b & 



da db 



Eliminating a we may also write for h a — : 



dx d.v 



L 



da i 

 da \ 3 d6 d# 



d.r ] 2 dxd'a 



( aV 



da d 2 a 



and as — must be positive, just as — , the sign depends on: 

 dx d# s 



2 d"a 3 db da 



3 dx 2 ~ 2 dx ~dx ' 



And if we put b 2 ^nb l> a = A -\- 2Bx -\- Cxi, this form becomes : 



_ C [I + (n-1) «] - (n-1) [B + C.*J 

 or 



| C + - (n-1) Cx - (n- 1) (5 + Cx) 



or 



2 C 2 B+Cx 



3w — l 3 3 



Now we have found above B -\- Cx = + I y^ -^ , and j: 



B = a l3 — a x and C=a x -\- a 2 — 2« M , the sign depends on: 



«1 + 0,-2^, 



2K 2 



n — 1 

 If this sign is positive we have the case treated first. So for this case 



(«1 -«!■>■— («11— ©l)^ , 



^> n — 1 



«ii — a i 

 or 



>n. 



is certainly necessary, but not sufficient. 



With the following numerical values the conditions necessary for 

 the first case, and the condition that the two values of x occur in 

 the diagram, are satisfied. 



