[Read before the Philosophical Institute of Canterbury, 1st September, 1897.]

[Abstract.]

In this paper, which is essentially mathematical, the author takes the expressions for the potential in the interior of solenoids as given by Maxwell and Thomson and Tait in terms of zonal spherical harmonics, and differentiates them with respect to “z,” the axial direction of the coil. As a result of this differentiation it is shown that this component of the magnetic force can be simply expressed in terms of zonal harmonics, and as far as n = 6 the relations

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d/dz(P_n(θ)/rⁿ⁺¹) = (n+1) P_n+1(θ)/rⁿ⁺²

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d/dz(rⁿP_n(θ) = -nr^n-1P_n-1(θ)

are proved. The general proofs of these identities with those of two more of a similar nature form the subject of another paper.

The solution of the main problem is divided into six cases, and general expressions given in four of these, from which the component parallel to the axis at any point can be obtained with the help of Perry's tables of zonal harmonics. The expressions for the magnetic force in the two remaining cases can be easily written down from the previous discussion.

From the general formulæ the following special cases are derived:—