194 ON THE DETERMINATION OF THE ATTRACTIONS 



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 F which will render the quantity (5) 

 a proper minimum. But it follows, from the principles of the 

 Calculus of Variations, that this function F , whatever it may 

 be, must moreover satisfy the equations (2') and (9). If then 

 there exists any other function V v which satisfies the last-named 

 equations, and the given values of V" and F/, it is easy to per- 

 ceive that the function 



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

 quantity A may be. Suppose therefore that A originally equal 

 to zero is augmented successively by the infinitely small incre- 

 ments BAj then the corresponding increment of F will be 



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

 value, however great A may become, seeing that by what pre- 

 cedes the variation of this quantity must be equal to zero what- 

 ever the variation of Fmay be, provided the foregoing conditions 

 are all satisfied. If then, besides V there exists another func- 

 tion F t satisfying them all, we might give to the partial dif- 

 ferentials 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 t 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). . 



5. Again, it is clear that the condition (4) is satisfied for 

 the whole of F 2 '; and it has before been observed (No. 2) that 

 when Fis determined by the formula (1), it may always be ex- 

 panded in a series of the form 



Hence the right side of the equation (9) is a quantity of the 

 order w" 1 '* 1 ; and u being evanescent, this equation will then 



