CuLVERWELL — SoluUons in the Calculus of Variations. 389 



2. If F{y) = if -\- "^ay-i and if the points and P^ be given, then, 

 whether the tangents at those points be given or not, the stationary- 

 solution gives a minimum for small variations of y, and large ones of y. 

 If, however, y also may have large variations, it is evident that the 

 stationary solution for fixed tangents could not, in general, be a 

 minimum. This, of course, follows either because Y^ must be zero 

 everywhere, as already proved, or at once from the condition that if 

 the tangent can have a quite arbitrary value throughout, it can have 

 an arbitrary value at the limit. It is easy to see in this case that the 

 stationary solution, when and P alone are given, does give a true 

 minimum to the integral when y and y are both quite arbitrary. 



3. If we apply the condition to the well-known evolute problem, 



where Fiif = (1 + y'^Yly-, we find that E can change sign when y 

 alone can have finite variations, thus showing that, not merely is the 

 cycloidal solution not a minimum, when we are allowed to use a series 

 of cycloidal or circular curves, as was long ago pointed out, but also 

 that it is not a minimum, even when the tangent is not permitted to 

 have a finite variation. 



§ 13. "When there are conditions, the criterion still holds in general 

 (but one must always be prei)ared for exceptional cases). Take first, 

 problems of relative minima. A single example will suffice to show 

 how the criterion is obtained for all integrals, whether single or 

 multiple. 



Let the problem be to find the curve of given length joining 

 and P, and enclosing the minimum area. 



Here, TI - 



yds, and the solution is got by making 

 f(y + Ajl +y'')dx 



stationary. 



There, referring to fig. 1, and following the previous method, 



m^I{AB)^ I{BP)-{AP)=^^^Q -y) dx^\^\y -y)dx ■ (23) 



but since the length is to be the same in both cases, 



X'^l +y'^dx+\ X-^l ^y'^dx = \ X Jv + y""- dx + \ Xjl+fdx. 



J A J B J A J B 



