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V2<£ = 0 ; that is, the electric forces obey the laws of electrostatics. Similarly a, b, c are derivatives of another function ^ satisfying the same equation. The only difference is that ^ may be multivalued. The magnetism is that due to steady electric currents. If several wires meet in a point, the total current is zero. This expresses itself in terms of a, 6, c as a relation between the " circulations." The method then consists in forming the solutions which apply to the parts at a distance on the two sides from the region of irregularity, and in accommodating them to one another by the conditions which hold good at the margins of this region in virtue of the fact that it is small.
In the application to the problem of fig. 3 we will suppose that the conductors are of revolution round 2, though this limitation is not really imposed by the method itself. The problem of the regular waves (whatever may be form of section) was considered in a former paper*. All the dependent variables expressing the electric conditions being proportional to in (4) compensates V2d2/dz2, so that
s2    dy2,
also R and c vanish.   In the present case we have for the negative side, where there is both a direct and a reflected wave,
P, Q, R = e^ (Hrf-™* + K^kz] (-j- , ~- , Q\ logr, .........(13)
^'               v                         ' \dx   dy     )    8                v    y
where r is the distance of any point from the axis of symmetry, and HJ, K^ are arbitrary constants.    Corresponding to (13),
V(a, b, c} = e^ (— H-,e-ikz + K-,eikz] (-7- ,--*-, 0 ) log r......(14)
\.di/      dec     j
In the region of regular waves on the positive side there is supposed to be no wave propagated in the negative direction. Here accordingly
P   O  R = ff pUpt—kz) (__    _    Olloo-r                     f]^
JT, v, j* — jiae *      M  ,   ,   ,   , v   logr, ..............(±0)
VW/vV        tu (/           J
*             t/        *
V (a, b, c) = ff^w-w (- ~ , •%-, 0 ) log r, ...........(16)
\   cLy   dx     )
H2 being another constant.   We have now to determine the relations between the constants -fi^, J£l} H2, hitherto arbitrary, in terms of the remaining data.
For this purpose consider cross-sections on the two sides both near the origin and yet within the regions of regular waves. The electric force as expressed in (13), (15) is purely radial. On the positive side its integral
* Phil. Mag. 1897, Vol. XLIV. p. 199;   Scientific Papers, Vol. iv. p. 327.d with 6.    The same meth< may be applied to the other formulas (9), (10), (11).