408 Mr green, ON THE DETERMINATION OF THE 



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.dx, dx,du=^^^^ (l-2r' ^') d^,dl,...dldh. 



• But since 1 - 2,'+' ^^ = v + ^»S,'+' ^, 



a; ' Ur 



the expression (5) after substitution will become 



fd^^d^i d^sdhui tti ih a.A""'!/""'"'. 



Applying now the method of integration by parts to the variation of 

 this quantity, by reduction, we get for the equivalent of (2) the equation 



^- dh^^ V" ,^ a;) hdh^^^ ^^''^ ar'dl' ^' "" ^^^a:-dlr' 

 + A^2^ X 2-^^ - A'22-Ml -^^ (2") 



where the finite integrals are all supposed taken from r = l to r = * + l, 

 and from r' = 1 to r' = * + 1. 



The last equation may be put under the abridged form, 



d^ . ( ^«:^ dV 



dJi 

 provided we have generally 



o = -^+(»-s5-)^ + vr (n. 



