CuLVERWELL — SoluUons in the Calculus of Variations. 385 



Thus, the right-hand side of (1 1) is reduced to the terms in {_F- F) 

 plus those in (16) ; in other words, to the function Edx of § 2. Hence 

 we may write 



I{OABP)- 1{0P)= EBx, 



showing that JEBx > is a necessary condition for a true rainimum 

 under the given conditions. 



It follows from this expression that, unless Bx is necessarily of 

 determinate sign, the integral cannot have a minimum of the assigned 

 character. 



§ 8. It has now to be shown that the condition EBx > is 

 mfficient. The following slight modification of the method given by 

 Zermelo is interesting, though somewhat longer than the general 

 method given in § 9. 



Let OBEAB CP be a variation from OP, the stationary curve, and 

 let the variations be all small, i.e. less than k, in the portions OB, EA, 

 and CP; but let the higher fluxions have large variations in BE and 

 A C, as pennitted by the conditions. 



Join AP by any curve which shall have all its fluxions small, i.e. 

 less than k, and shall have contact of the proper order with EA or AB 

 at A, and with OP at P, so that OBEAP is an admissible variation. 

 Similarly, draw BP, so that ABB is part of an admissible variation. 



Then it has just been shown that 



li^AB) + li^BP) - I{AP) = j^EBx + (Z;)2 = j^ E T x + (Jc\. 

 Similarly 



I{BC) + 7(CP) - I{BP) =1^1 ^ + (% 



