L916]
APPROXIMATE SPHERES AND CYLINDERS
387
jo that In vanishes unless n = Zp. But L^ disappears in (18), presenting .tself only in association with C^p, which we are supposing not to occur. the last integral in (18) makes no contribution, reducing to
vanishes. Thus
= 2 log (*,0Z>)-
(21)
jhe same as in the former approximation, as indeed might have been anticipated, since a change in the sign of Cp amounts only to a shift in the lirection from which Q is measured.
The corresponding problem for the approximate sphere, to which we now -jroceed, is simpler in some respects, though not in others. In the general jase u, or r~l, is a function of the two angular polar coordinates 0, o>, and jhe expansion of Su is in Laplace's functions. When there is symmetry ibout the axis, w disappears and the expansion involves merely the Legendre Junctions Pn (/u), in which p, = cos d. Then
ivhere Glt (72, ... are to be regarded as small. We will assume 814 to be >f this form, though the restriction to symmetry makes no practical difference .n the solution so far as the second order of small quantities.
For the form of the potential (<£) outside the surface, we have
(23)
+ Bu |#o + 2«0 Hl Pa + 3 W02 //2 Pa + . . . }
+ (Sw)2 [H.P, + 3w0 HZP. + . . . + ij> (p + 1) uf-*HpPf], . . .(24)
.n which we are to substitute the values of Bu, (&u)* from (22). In this equation ^ is constant, and Hl} Hz, ... are small in comparison with #„.
The procedure corresponds closely with that already adopted for the cylinder. We multiply (24) by Pn, where n is a positive integer, and integrate with respect to p over angular space, i.e. between — 1 and + 1. Thus, >mitting the terms of the second order, we get
u0-Hn = -H0Cn, ............ • ............... (25)
is a first approximation to the value of Hn."
25—2ularity and of the transmitted wave on the right are represented