Equations in a Homogeneous Isotropic Medium. 37 



and therefore 



<^(VV(^)= - PP/(a)^(-m2) cos m(^-a)c;m^a; (20) 

 applying wliich to (17) we obtain 



E= — I I ^/7i tZa Eq cos m(^—a)[ cosh .sinhj^-^ 



Ho . . . sinh qt~\ 



-\ m sin miz—a) — . 



c ^ ' q J 



H= j I dmdal Hq cos m(5 — a)( cosh + - sinh J g^ 



Eo . , X sinh of] 



H -msmmiz—a) \. 



a ^ ' q J ■ 



(-•(^l) 



Eo . , X sinh of 



—^msmmiz—a) 



fj, ^ ' q 



in which, by (16), 



q^ = a''-mH^, (22) 



and Eo, Hq are to be expressed as functions of a, whilst E and 

 H belong to z. Discontinuities are now attackable. 



The integrations with respect to m may be effected. In 

 fact, I have done it in three different ways. First by finding 

 the effect produced by impressed force. Secondly, by an 

 analogous method applied to (17), transforming the differen- 

 tiations to integrations. Thirdly, by direct integration of 

 (21) ; this is the most difficult of all. The first method was 

 given in a recent paper* ; a short statement of the other two 

 methods follows. 



11. Transformation of (17). — In (17) we naturally consider 

 the functions of qt to be expanded in rising powers of q^, and 

 therefore of V^^ leading to dijft'erentiations to be performed 

 upon the initial states. But if we expand them in descending 

 powers of Vj we substitute integrations, and can apply them 

 to discontinuous initial distribution. 



The following are the expansions required : — 



63' 



9 



* " Electromagnetic Waves," Phil. Mag. October 1888. 



