## Extract

This note refers to the paper ‘An electromagnetic interpretation problem for the sphere’ (Slichter 1952), which will be designated as ‘ref. (1)’. In that paper it was assumed that a sphere of unknown electrical properties varying only in the radial direction was subjected to electromagnetic induction from an external concentric current system. From observations of the magnetic vector on the sphere’s surface the variation of the electrical properties with depth was determined. One of us (A.T.P.) has called attention to special excitation functions for which the proposed problem has no unique solution. The reason for the failure of the general method in these cases is easy to see. As developed in ref. (1), the interpretation process depends upon the evaluation of the coefficients in the asymptotic expansion of a so-called ‘kernel function’, *K*(*v*), in inverse powers of *v*. Here *v* is the separation parameter for the partial differential equation of the problem. The solutions in ref. (1) for the two components of the magnetic vector, equations (21) and (23), and the expression for the distribution of source current, *m*(*ϴ*), equation (25), take the form of infinite integrals with respect to *v, 0≤ v ≤∞*, where the integrands contain products of the normal functions of the problem, *T _{v-1/2}* (cosӨ) and

*R*(

*p, v*), or of their derivatives. (Here

*T*(cos

_{v-1/2}*Ө*) is the Legendre function, of degree

*v*–1/2, and

*R*(

*p, v*) is a function only of

*p*and

*v*.) However, when the factor in the excitation function which is dependent upon the co-latitude,

*Ө*, is itself a normal function for the problem, and thus expressible by a single value of

*v*, say

*v*

_{1}, then the kernel

*K*(

*v*) degenerates to a constant and the required power series expansion does not exist. It is obvious that the normal functions correspond to physically possible modes of excitation. For example (see equations (21) and (25)), when

*v*=

*v*

_{1}both

*H*and

_{Ө}*m*(

*Ө*) as functions of

*Ө*are proportional to

*T*

^{1}

_{v1-1/2}(cos

*Ө*) = (1/4-

*v*2/1)

^{-1}

*T*

^{-1}

_{v1-1/2}(cos

*Ө*). This function satisfies the physical condition that

*H*and

_{Ө}*m*(

*Ө*) vanish at

*Ө*= 0, and

*Ө*=

*π*-

*є*, where

*є*is arbitrarily small.

## Footnotes

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- Received September 25, 1952.

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