326 Mr PocMington, Electrical Oscillations in Wires. [Oct. 25, 



Suppose now that we shift the curve formed by the Hertzian 

 elements through a small distance of the second order. The 

 effect is to change the value of the component tangential to the 

 cross-section by a finite amount which varies as the cosine of 

 some azimuth angle, and that parallel to the axis of the wire by 

 an amount of the first order of small quantities. 



By making such a shift of appropriate magnitude and direction 

 at every point, we can therefore eliminate the component tangential 

 to the cross-section. At the same time, the component parallel 

 to the wire is unaltered (to our order). Hence we may still 

 derive the surface condition from (1) by taking the integration 

 along the axis of the wire. 



The condition thus obtained is 

 = ^jds^U + a 2 {I jdslXU + mfdsmXIl + njdsnXTi} . . .(2). 



5. Circular Ring. The simplest case that we can consider is 

 that of a circular ring. Let the radius of the ring be a, the 

 radius of the wire e as before, and let the axis of symmetry be 

 chosen as the axis of z. We shall assume \ = A cos r<£, where <j> 

 defines a point on the axis of the wire. This assumption will be 

 justified later. 



At the point (vr, 0, z) 



f dX f 2n 



I ds -j- n = — r A I rf(/>n sin rcf> 



= - rA I d<j> n o (sin rd cos r<f> + cos rd sin r</>) e lpt , 



where n o is the value that n takes when <f> is put for (0 — <£) and 

 for t, and is thus a function of </>, zr, z only, 



= — 2rAe ipt sin rd \ d<f>TI cos r<p ; 



IdslXll = — A I ad(f>U cos</> cosr<j> 



= Aae ipt cos (r + 1)0 1 d<f)U cos(r + l)</> 

 Jo 



+ cos (r— 1)0 1 d(f)~n. cos (r — 1) <f> 



Jo 



as above ; and JdsnX U = 0. 



