between Confocal Elliptic Cylinders. 229 



cases. The case of waves along z, already treated, will serve 

 as the first. The second must be obtained by arbitrarily 

 making two components o£ magnetic force vanish. On 

 addition of the resulting solution to that obtained above, the 

 general values of the forces for all possible oscillations may 

 be found. 



The case (b, n) = cannot be made to satisfy the surface 

 conditions ; therefore we are reduced to the case 



a=0, b = 0, . .- . . . . (10) 

 which* involves, of necessity, 



Z = (11) 



By equation (4), 



where <£ does not contain z. 



Hence c is not a function of z (12) 



Substituting in the circuital relations, it is found that c 

 must satisfy the differential equation 



^P(A)^v'P^)| + v/-Pw| i v/"PH|-; + J(X-^) c =0. (13) 



Since Z = 0, and the electric and magnetic forces are per- 

 pendicular, both surface conditions are included in Y = 0; 



^- =0 at the surfaces (14) 



Let c=LMcos (&V£ + e), where L is a function of A only, 

 and M of ju. 



1 ?3 2 L _ L 1 d 2 M _ I 

 where 



L9^ + Kj? = "I (x "' ,) " • • (15) 



-S 



xm- HtSw • • <IS > 



We may therefore take 



L da 2 4 ' ( Q7) 



■1 d 2 M = -6 + k 2 v 

 MdfP 4 



where 6 is an arbitrary constant. 



The first of these equations will now be considered. 



