410 Sir W. Thomson. On Electrostatic Screening [Apr. 



/>(m+n), for z = + 00; and p(mn), for z = oo. -..(!{ 



If, in (4) with nz instead of + mD, we put 



we find, as the equation of the equipotential surfaces, 



2 cos ma + <*+ c-~ = Ce-" ( 



By taking n = 0, or n = m, we fall back on the cases of 5-8, 

 9, 10, respectively. 



12. To find an approximate equation for the equipotentials at dii 

 tances around primitive bars small in comparison with a, the distanc 

 from bar to bar, let x and z be so small that we may neglect 

 powers of mx, mz, and nz, above the square, which implies that C 

 small of the same order as (mx) 2 and (mz) 2 , (11) becomes 



where r 2 denotes m~~ 2 C. 



This shows that, to the degree of approximation in which we negl 

 cubes and higher powers of mx, mz, nz, the equipotential is a row 

 elliptic cylinders of eccentricity nrj v/2, with their greater diamet 

 planes perpendicular to the plane of the primitive bars. When n = 0, 

 the equipotential is a row of circular cylinders having the primitive 

 bars for their axes ; and this is true to the higher approximation in 

 which we need only neglect powers above the cubes of mx and mz, as 

 we see by going back to equation (17), with n = 0. 



13. The conclusions of 12 are useful for detailed investigation of 

 the screening effect of plane gratings of circular or elliptic, straight 

 parallel bars electrified with given quantities of electricity and placed 

 with their planes perpendicular to the lines of force in a uniform 

 field of force, and to corresponding problems in which potentials are 

 given, as in Maxwell's 203 205. 



14. Instead of a single row of parallel equidistant infinitely thin 

 bars in one plane, let us take for primitives two or more such rows, 

 parallel or not parallel, all in one plane or not in one plane. We may 

 thus form an endless variety of force-systems available for illustrating 

 or helping to solve problems which may occur. Towards the several 

 problems of electric screening we find important contributions by 

 considering in two parallel planes rows of primitive lines parallel to 

 one another for one case and perpendicular to one another for another 

 case. The consideration of three rows of primitive lines in one plane, 

 dividing it into equal and similar triangles alternately oriented in 



