Art. LII.—Two Spherical Harmonic Relations.

ByG. Coleridge Farr, D.Sc.

[Read before the Philosophical Institute of Canterbury, 3rd September, 1902.]

The relations proved in this paper were given without distinct proof as following simply from the generalised form of four others which I discovered in working out the expressions^* for the intensity of the magnetic force in the interior of coils of various lengths. They were, however, cut out by Professor Lamb, who very kindly communicated my paper to the Royal Society, as he did not see how they were obtained. This of itself causes me to think they may be new, and, as the original four are, I believe, new ones, hope in this direction is strengthened. I have known them for some years now, but as they follow simply from the four already published I had not up till now thought them worth publication. I have, however, been recently rather strongly advised to print them.

Using the notation of my previous paper, the original four are these:—

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(1) d/dx Pσ/rσ+1 = 1/rσ+2d/dθ Pσ + 1

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(2) d/dxrσPσ = rσ-1d/dθ Pσ − 1

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(3) dι/dzι Pσ/rσ+1 = (σ + ι)!/σ! Pσ + i/rσ+ι+1

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(4) di/dzιrσ Pσ = (− 1)^ι σ!/(σ − ι)! rσ-i Pσ − i

Now, since the order of differentiation is indifferent, we have

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dι/dzι/d/dx Pσ/rσ+1 = d/dzx/dι/dzι Pσ/rσ+1 (5)

Taking first the left-hand side of the equation and making use of (1) we have

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di/dzι/d/dx Pσ/rσ+1 = di/dzι1/rσ+2 d/dθ Pσ + 1 (6)

Now, taking the right-hand side of (5) and using (3), it follows

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d/dx/dι/dzi Pσ/rσ+1 = d/dx (σ + ι)!/σ! Pσ + i/rσ+ι+1
= (σ + ι)!/σι 1/rσ+ι+2 d/dθ Pσ + i + 1