200 ON THE DETERMINATION OF THE ATTRACTIONS 



belonging to the extreme limit before marked with two accents, 

 by simply assigning to h an infinite value. 



The easiest way of transforming the equation (2) will be to 

 remark, that it is the general one which presents itself when we 

 apply the Calculus of Variations to the quantity (5), in order to 

 render it a minimum. We have therefore in the first place 



and by the ordinary formula for the transformation of multiple 

 integrals, 



. . . dx t du = 





But since 1 - 2/ +1 -Hp- = ^ + #2/" 

 the expression (5) after substitution will become 



)(,/ + w, 



(V 



' - v s,-* 



*/ 



Applying now the method of integration by parts to tlie varia- 

 tion of this quantity, by reduction, we get for the equivalent of 

 (2) the equation 



where the finite integrals are all supposed taken from r = 1 to 

 r = s + 1, and from r'= 1 to r s -f 1. 



