IX. On the effect produced on the deviations of the compass by the length and arrangement of the compass-needles; and on a new mode of correcting the quadrantal deviation

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Abstract
In the mathematical investigations of the deviation of the compass which have been hitherto published, and in the practical methods for correcting the deviation which have been proposed by Captain Flinders, Mr. Barlow, and Mr. Airy, the assumption is tacitly or expressly made that the length of the compass-needle may be considered as infinitesimal compared with the distance of the nearest disturbing iron. Expressed mathematically, as only even powers of the ratio of the projection of the needle on the direction of the disturbing iron to the distance of the disturbing iron enter into the expressions, the assumption is that the square and higher powers of that ratio may be neglected. By this assumption the formulæ which express the deviation are materially simplified; and on the supposition that the iron of a ship consists entirely of iron of one or other of the two extreme qualities described magnetically as “hard” iron and “soft” iron, the following expression is accurately true:— sin δ = A cos δ + B sin ζ' + C cos ζ' + D sin ( ζ + ζ' ) + E cos ( ζ + ζ' ); in which ζ is the azimuth of the ship’s head measured eastward from the correct magnetic north ; ζ' is the same azimuth, but measured from the direction of the disturbed needle ; δ = ζ-ζ' is the easterly deviation of the needle; A, D, E are coefficients depending only on the distribution of the soft iron of the ship, and being independent of the magnetic dip and force, and therefore not changing with a change in the geographical position of the ship. B and C are coefficients depending partly on the distribution of the hard and soft iron of the ship, and partly on the dip and horizontal force, and therefore changing with a change of geographical position of the ship, as well as with a change in the magnetism of the hard iron in the ship.

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