328 Mr.Cankrien on the Problems of the Calculus of Variations. 

 Now, fd x 2 B uu' = fd x B ~ = B # -fd x« J B ', and 

 therefore^ dx2Buu' = — J dxi? B'. 



Similarly / ^2E«'«"= — jr rfiEi/ 2 . 



AgaiD,/tf#2DW = D ^ -/MD« 2 - DV 

 + fdxu* D", and therefore / S^ 2D u a" = /£? « 2 D"- 



^2Dm' j ; and so on. Hence 



W,- W., = U,-U„ + ^fdx {(A-B' + D') u* + 



(C-2D-E)?r- + Fu" 2 + &c.} + &c. 



Now, in order that U,— U„ may be a maximum or a mini- 

 mum, the term multiplied by k- in the above expression must 

 be either positive or negative whatever be the function u; and 

 U — U , will be a maximum or a minimum according as this 

 term is negative or positive. It is not easy to find generally 

 the relation of the coefficients of ?r, u '% 71"*, &c. to one an- 

 other by which these conditions are fulfilled. In order to de- 

 termine it in particular cases we may remark, that if the values 

 of <P' (x) be found corresponding to the series of values x y/ , 



x / + //, x lt + 2 h, &c. x f/ + n — J . k 1 h being = — — - and n 



a large number; then <P (x f ) — <P (x ) is positive or negative, 

 according as the sum of the positive values of <P' (x) found in 

 the manner just mentioned, is greater or less than the sum of 

 the negative values. If all these values of <P (#) are positive, 

 or all negative, then (.r y ) — <P (x / ) is in the one case posi- 

 tive, and in the other negative, as is evident. If then we de- 

 duce the values of A, B, C, D, E, F, &c. from V, and the ex- 

 pression (A-B' + D") u- + (C- 2D-E') u'*+ Fm" 8 + &c, 

 which we will call <P' (x), be reduced as much as possible by 

 means of the equation N— P + Q"— R'" + &c. = 0, we shall in 

 many cases see whether it be possible so to assume the function 



u that —-- J dx <P' (x) shall be either positive or negative: if 



this be possible, then the equation N— P'+Q '— R"+ &c. 



= does not make fd x V either a maximum or a mini- 

 // mum: 



