Resistance of Thin Metallic Films. 495 



over a sufficiently large number of groups, bat in the general 

 case the distribution of the field inside the groups, necessary 

 to satisfy (2) and (3), is extremely complicated. There is 

 one particular case, however, in which we can obtain a 

 solution, and we can, I think, see by arguing from this case 

 that the solution for the general case cannot be far different 

 from this. 



Let us consider the case where the bounding gaps form 

 the faces of hexagonal solids filling up the space, and suppose 

 the field to act parallel to the line MP (fig. 8) which is per- 

 pendicular to one set oi faces. Then a uniform field through- 

 out the system will satisfy all the conditions. For suppose 

 that we assume such a constant field Xj. In order to satisfy 

 condition (3) the value of V at any plane will be 



V=^F(a)X 1 cos7, . 

 o 



where y is the angle between the normal to this plane 

 and MP. 



This adjustment of V will satisfy (2) since we shall have 



A V B =JxF(a)X 1 cos(0) J 

 A V + C V B = | XF(a) X, { cos | + cos J- J . 



If we now adjust the absolute value of Xx so that X = X : — j 



where V is the potential drop across one of the gaps perpen- 

 dicular to MP, and / is the distance between an adjacent 

 pair of such gaps, (1) will be satisfied along the line MP, 

 and it will also be satisfied along such a line as TP, since, 

 although this line intersects twice the number of gaps per 

 centimetre as the line MP, the potential drop across each of 



7T v" 



them is only V cos — = — . 

 J 6 2 



Thus we see that the solution for this particular case is 

 exactly the same as if the gaps were all perpendicular to the 

 field X and separated by distances /, i. e. the solution is exactly 

 the same as for the problem we have already solved in this 

 paper, and the specific resistance is given by the expression 

 in equation (10). 



If the field is not perpendicular to a set of bounding gaps 



