170 Mr Mayall, On Ow^rent Sheets, especially [Feb, 26, 



(see Heine, Kugelfunctionen), where 



{o-df = cosh a — cos 6. 



Let o- = (r^^ be the equation of the current-sheet and suppose 

 the external magnetic field to be symmetrical about the axis 

 of the ring, and therefore HqXI and ^ independent of t/t. Then 

 denoting the specific resistance in this case by p, equation (3) 

 becomes 



d . , d^ k sinh ad d .^ „ . 



d d^ kip d /n , n\ /«)k\ 



if O, n and O^ vary as e'-^*. 



The type of function which satisfies Laplace's equation for 



the co-ordinates here used is (ad) P^ (cosh a) sin nd where 



2w -h 1 

 P„ (cosh a) is a zonal spherical harmonic of degree — ^ — with 



ia as argument ; its properties as well as those of the allied 

 function Q„ (cosh a) are fully investigated by Dr Hicks in the 

 paper to which allusion was made above. 



If however we attempt to solve (25) in the same way as 

 the previous cases, viz. by assuming O^ to be of the form 



Aq (ad) Q^ (cosh a) sin nO, 



Xlj of the form A^ (ad) P^ (cosh a) sin nd, 



and Og of the form 



A^(ad) Q^^ (cosh a) sin nd, 



we find that these values of Oj and fl^ cannot represent the 

 potential due to the current sheet, for their differential coefficients 

 with respect to a are 



[I sinh aPJ(ad) + (ad) PJ^ A, sin nd, 



and j 2 ^^^^^ ^QJC^"^) + (o"^) Q„'[ ^2 si^^ »*^. 



and these cannot be equal when a = a^^ for all values of d unless 

 A^ and A^ are both zero. 



The point in which this case differs from those already dis- 

 cussed lies chiefly in the fact that the expression for the typical 

 harmonic instead of being the product of three terms each of 

 which is a function of one co-ordinate only, contains in addition to 

 these a term which is a function of two co-ordinates a, d. 



