Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/104

In these equations a, b, c, d, e, f, g, h, k are nine constant coefficients depending on the amount, the arrangement, and the capacity for induction of the soft iron of the ship.

P, Q, and R are constant quantities depending on the permanent magnetization of the ship.

It is evident that these equations are sufficiently general if magnetic induction is a linear function of magnetic force, for they are neither more nor less than the most general expression of a vector as a linear function of another vector.

It may also be shewn that they are not too general, for, by a proper arrangement of iron, any one of the coefficients may be made to vary independently of the others.

Thus, a long thin rod of iron under the action of a longitudinal magnetic force acquires poles, the strength of each of which is numerically equal to the cross section of the rod multiplied by the magnetizing force and by the coefficient of induced magnetization. A magnetic force transverse to the rod produces a much feebler magnetization, the effect of which is almost insensible at a distance of a few diameters.

If a long iron rod be placed fore and aft with one end at a distance x from the compass needle, measured towards the ship's head, then, if the section of the rod is A, and its coefficient of magnetization κ, the strength of the pole will be AκX, and, if $A={\frac {ax^{2}}{\kappa }}$, the force exerted by this pole on the compass needle will be aX. The rod may be supposed so long that the effect of the other pole on the compass may be neglected.

We have thus obtained the means of giving any required value to the coefficient a.

If we place another rod of section B with one extremity at the same point, distant x from the compass toward the head of the vessel, and extending to starboard to such a distance that the distant pole produces no sensible effect on the compass, the disturbing force due to this rod will be in the direction of x, and equal to ${\frac {B\kappa Y}{x^{2}}}$, or if $B={\frac {bx^{2}}{\kappa }}$, the force will be bY.

This rod therefore introduces the coefficient b.

A third rod extending downwards from the same point will introduce the coefficient c.

The coefficients d, e, f may be produced by three rods extending to head, to starboard, and downward from a point to starboard of