228 Mr. Nicholson on Electrical Vibrations 



Since <£ cannot contain \, this gives the two equations 



Hence 



<b = — 7 — - — cos (kz + a) cos (kYt + e) ; 



v 7 - p W 



whence the components of electric force are 



X = / A . cos (kz + a) . cos (kYt + e) , Y = Z = a.. (8) 



By equation (6) the first surface condition is satisfied ; and 

 since (Y, Z) both vanish, so also is the second. 



This solution represents the simple case of waves propagated 

 along z; and we note that the force is independent of the 

 size of the boundaries. It holds also inside a single cylinder. 



X is nowhere infinite, for A = a* only on the imaginary curve 



The magnetic force at any point is normal to the confocal 

 hyperbola through it, and has the value 



A 



b=- ^ , = . sin {kz + a) , sin (kYt + e). . (9) 



It is readily seen that this is the only motion of the kind ; 

 for if we had taken (X, Z) = 0, we should have found 



Y= - 7 --=-cos(£~ + /3)cos(&V£ + <?), 



which cannot vanish at the conductors unless B = 0. 



Again, if (X, Y) =0, Z is a mere trigonometric function of 

 z and t, and cannot vanish at the conductors unless it does so 

 everywhere. 



Finally, when waves are propagated along the axis, the 

 electric and magnetic forces are normal to the confocal elliptic 

 and hyperbolic cylinders respectively, and are of magnitudes 

 given by (8) and (9). 



General Solution. 



It is readily seen from the equations and surface conditions 

 that the general solution can be built up from two particular 



