H 



362 Mr. 0. Heaviside on Electromagnetic Waves, and the 



the rest expresses the second wave, reaching r when t = (r + a) /v. 

 After that the actual H is their sum, viz. 



H=^ e -p^fl + ^r cosh _JiL sinh |^ . (210) 

 fjivr \ prj L pa J v 



agreeing with (206), when we give q therein the special 

 value p/v at present concerned, and hi—Anrh. 

 At the front of the first wave we have 



E. = €-<> t Jva/2fjLvr, . . . . . (211) 



so that the energy in the travelling shell still subsides to zero. 

 Equation (211) also expresses H at the front of the inward 

 wave, both before and after reaching the centre of the sphere. 

 The exchange of a and r in the [] in (209) produces the 

 corresponding internal solution. 



35. Unequal p 1 and p 2 . General case. — If we put djdr=SJ, 

 we may write (206) thus, 



- E | [(V- ^)(V + ^— > + (v- JXV- %r+~y M 



It is, therefore, sufficient to find 



*-«(■— y-y, (213) 



to obtain the complete solution of (212) ; namely, by per- 

 forming upon the solution (213) the differentiations V and 

 the operation k x . This refers to the first half of (212) ; the 

 second half only requires the sign of a to be changed in the []. 

 Now (213) is the same as 



Expand the two functions of p in descending powers of p, 

 thus, 



r-^ =e -^\i + ^ + ^ h+ ..], . . .( 2 i6) 



where the h's are functions of r, but not of p. Multiplying 

 these together, we convert (213) or (214) to 



^.-f^tl + lh + ^ + .-.JC/n • (217) 



where the i's are functions of r, but not of p. The integra- 

 tions can now be effected. Let/ be constant, first. Then 



