402 Mr green, ON THE DETERMINATION OF THE 



From the nature pf the question de minimo just resolved, there can 

 be little doubt but that the equations (2') and (9) will suffice for the 

 complete determination of V, where V" and V-l are both given. But 

 as the truth of this will be of consequence in what follows, we will, 

 before proceeding farther, give a demonstration of it; and the more 

 wiUingly because it is simple and very general. 



4. Now since in the expression (5) u is always positive, every one 

 of the elements of this expression will therefore be positive; and as 

 moreover V" and F"/ are given, there must necessarily exist a function 

 Fo which will render the quantity (5) a proper minimum. But it 

 follows, from the principles of the Calculus of Variations, that this 

 function Va, whatever it may be, must moreover satisfy the equations 

 (2') and (9). If then there exists any other function F", which satisfies 

 the last-named equations, and the given values of V" and V^, it is easy 

 to perceive that the function 



will do so likewise, whatever the value of the arbitrary constant quan- 

 tity A may be. Suppose therefore that A originally equal to zero 

 is augmented successively by the infinitely small increments SA, then 

 the corresponding increment of V will be 



Sr={F,-V,)SA,' 



and the quantity (5) will remain constantly equal to its minimum 

 value, however great A may become, seeing that by what precedes 

 the variation of this quantity must be equal to zero whatever the 

 variation of V may be, provided the foregoing conditions are all satis- 

 fied. If then, besides F"o . there exists another function F"; satisfying 

 them all, we might give to the partial differentials of F", any values 

 however great, by augmenting the quantity A sufficiently, and thus 

 cause the quantity (5) to exceed any finite positive one, contrary to 

 what has just been proved. Hence no such value as F, exists. 



We thus see that when F"" and F"/ are both given, there is one 

 and only one way of satisfying simultaneously the partial differential 

 equation (2), and the condition (9). 



