GREEN^ FORMULA. 23 



Integrating this expression 5y parts, we have 



dx = U dydz- \-d 



~ " " ~ 



The first term of the second member should be extended to the 

 whole surface S, and the second to the volume bounded by this 

 surface. Repeating this operation for the other co-ordinates, the 

 sum of the integrals relative to the surface S will be 



dydz-\ -- dzdx-\ -- dxdy ). 



x "^ "N. "X -^ I 



ox oy 02 



Now, if we consider V as a potential, which does not restrict the 



<)V 

 general character of the demonstration, the expression -- dydz, or 



, represents the flow of force through the surface element dydz; 

 that is to say, the projection of a surface element d$ on a plane 

 perpendicular to the axis x. It is the same for other terms, so that, 

 except for the sign, the parenthesis represents the excess of force 

 which traverses this element of surface. This parenthesis is thus equal 



av 



to - F n <afS, or ^S, and we have, finally, Green's formula : 



C f 



UAW?;= 



J J 



TTc 



(12) UAW?;= U </S- 



uav ww wav 



. + -- + . I 



Making U = i, we obtain the preceding equation (u). 

 When the functions U and V are identical, we have 



fa*- fv 



J J ^ 





If the function V represents the potential of an electrical system, 

 the force F is determined by the equation, 



3* 



which gives 



(14) 



// f* 



VAV^= V ^S - PV0, 

 ~bn 

 / */ 



