^9ê 



On the cojitrary, it' v„ is given, v lias another value for the 

 surround ing' elements, than if r„ = 0. Be in the element at xyz 



v.rj/z=9{'i', y,^,v^dv^) (4) 



and let us try to determine the function g, the function ƒ being 

 given. 



Now take the mean of formula (3), a fixed value I'l being ascribed 

 to V in a certain element dx-^dy^ dz-^. 

 In X, y, z, according to (4) 



Vxi,z—9{iv—''Vi,y—y^,z—z^,v^dx^dy^dz;). ... (5) 

 For the first member we therefore get 



9{^h,yv -1' v.dx^dy^dz,) 

 as ƒ and y do not depend on the direction of the line joining the 

 elements. In the integral, (5j cannot be applied to the element 

 dx^ dy^ dz^ ; however, this element gives 



v,fix„y,,z,)d.x^dy,dz^ 



Further taking g {0, 0, 0) zero, as it may arbitrarily be chosen, 

 we get 



J 00 



g{x^,y,,z,,v^dx,dy,dz^) = \ \ jg{.v-.v„y-y,,z-z,,v,diV,dy,dz,)f{a,'yz)dxdydz-{- 



00 



+ rj'{^,,y,,z,) dx,dy,dz,. 



This is true for all values of v^ dx^ dy^ dz^, hence g must contain 

 this quantity as a factor, and we obtain 



g{'V,,y,,z^) — j I L(.f — ,/;,, y-y„ z—z^)f{xyz)d.vdydz=f{a;,,y,.z^) 



00 



Now put X — x^ =1 §, y — z/, = 1], z — z^ = S, and omit the index, 

 then for g we get the integral equation 



g{'^,yA -JjT/^'^'-^-^' ^ + '^' ~^ + ^)^^^(^'^i^) dld,id^^j\xyz\ . (6) 



— » 



For g we have 



Vxyz — y{xyz) \\dv^ (7) 



from which it appears immediately that 



Vxyz Vo= .9(^2/4 »^o' ^ï'o (^) 



continuous. The integral-formulae obtained in this way are easier to deal with 

 mathematically, and besides the integral equation (6) has been solved, this being 

 not so easily found from the analogous sum -formula. 



