236 Electrical Vibrations between Confocal Elliptic Cylinders. 



becomes closer as the successive roots are taken in an in- 

 creasing sequence. 



In the particular case of short waves inside a single elliptic 

 cylinder, the forces must be finite at A = 0. Hence in 

 equation (43), 



€l =-kb. + J, 



and the a* term must similarly be finite. 



.'. c= -^-r. sin (k*/¥+\-B) sinh (kJW+v-kb) . e* v *, (50) 



is the approximate magnetic force for waves such as those of 

 light. The electric force may be deduced at once from this 

 expression. 



The period equation for a single cylinder defined by X is 



d ( sink\Q>*+\)t-b\ \ 



where only the large values of h are chosen, the approximation 

 being closer as they increase. 



A similar investigation for the case of long waves appears 

 impracticable. The waves peculiar to the cylinders cannot 

 have a length sufficiently great to ensure the utility of a 

 continued approximation to the solution of the equation. The 

 assumption of the existence of very long waves, when tested, 

 is found, as was perhaps to be anticipated, to lead to impos- 

 sible values. The approximate solution given above for short 

 waves is of remarkable simplicity, and almost quite accurate 

 in the case of light. 



A certain interest attaches to the class of waves worked out 

 in terms of elliptic functions ; but their period equation is so 

 complicated that it cannot readily lead to any numerical calcu- 

 lations, to which, on the contrary, the short periods readily 

 lend themselves. 



The most general type of oscillation possible in the space 

 consists of the combination of — 



(1) Waves of the doubly infinite series of possible periods 

 travelling round the axis of z. 



(2) Waves of any period whatever travelling along z. 



In this general vibration, the path of a ray is a spiral 

 curve round the axis of the cylinders, and the following 

 results will hold : — 



(1) The magnetic force never has a component normal to 

 the confocal elliptic cylinder through any point. 



(2) The electric force never has a component along the axis 

 of the cylinders. 



