CuLVERWELL — SoluUoiis ill the Calculus of Variations. 379 



Let also the function JE be defined as follows : — 

 U^F(x„ X,..., yi, 2/2. . -yi*''^---', &c., ^i(«^---), &c.) 

 - F(x„ X2..., yj, 2/2, . . . tji^'-'---\ &c., y(<^P---), &c.) 



-:^Y„,Qi-'^'---)-yi-'^'--^)^F-F-%, (2) 



where y is the value of y corresponding to the varied curve. 



§ 2. Then the necessary and sufficient conditions that the stationary 

 solution should be a true minimum compared with an integral obtained 

 from it by a permissible variation are 



Ta^.,.^0{a>a, P>b,&c.), (8) 



and F>0. (4) 



These conditions, (3) and (4), are to hold throughout the whole 

 extent of the integration when yi, ya have the values derived from 

 the stationary solution, while the fluxions of yi, yo, &c., have any finite 

 arbitrary values. 



§ 3. It will be supposed that the increment of every independent 

 variable is positive throughout the integration unless otherwise stated. 

 In case any independent variable should not fulfil this condition in the 

 solution of any problem, it will only be necessary to take a new inde- 

 pendent variable which does satisfy it, treating the old independent 

 variable as a new dependent one. Thus, if we are treating of a plane 

 curve where the independent variable x changes sign, either in the 

 original or varied curve, it is only necessary to introduce a new vari- 

 able s, to treat x and y as functions of s, and to apply the criteria in 

 the form in which they are given for three variables, instead of that 

 for two variables. 



It may sometimes, however, be well to consider the integral in its 

 original form, in which dx does change sign. 



§ 4. When it is said that the limiting variations are to be zero, 

 what is meant is^ that 



gy(r, ^, . . .) = 0, r <a, s <b, &c. (5) 



at every limiting point. This is a more general condition than the 

 ordinary one of "fixed limits," because here Sy(«.^. •••>, &c., may have 

 any finite values at the limits, a modification which immediately 

 follows from the condition that these quantities may have finite values 

 anywhere in the range of integration. 



§ 5. Thus the conditions of a permissible variation are — 

 («) Every variation typified by Syl''. ■?. • • •) must be continuous 

 throughout the whole extent of integration, and be zero at every 

 limiting point. 



