with the magnetic force exerted upon it that a small displacement of the centre of inertia by unequal dilatation, &c. produces a greater effect on the position of the magnet than a considerable change of the magnetic force.
Hence the measurement of the vertical force, or the comparison of the vertical and the horizontal forces, is the least perfect part of the system of magnetic measurements.
The vertical part of the magnetic force is generally deduced from the horizontal force by determining the direction of the total force.
If be the angle which the total force makes with its horizontal component, is called the magnetic Dip or Inclination, and if is the horizontal force already found, then the vertical force is , and the total force is .
The magnetic dip is found by means of the Dip Needle.
The theoretical dip-needle is a magnet with an axis which passes through its centre of inertia perpendicular to the magnetic axis of the needle. The ends of this axis are made in the form of cylinders of small radius, the axes of which are coincident with the line passing through the centre of inertia. These cylindrical ends rest on two horizontal planes and are free to roll on them.
When the axis is placed magnetic east and west, the needle is free to rotate in the plane of the magnetic meridian, and if the instrument is in perfect adjustment, the magnetic axis will set itself in the direction of the total magnetic force.
It is, however, practically impossible to adjust a dip-needle so that its weight does not influence its position of equilibrium, because its centre of inertia, even if originally in the line joining the centres of the rolling sections of the cylindrical ends, will cease to be in this line when the needle is imperceptibly bent or unequally expanded. Besides, the determination of the true centre of inertia of a magnet is a very difficult operation, owing to the interference of the magnetic force with that of gravity.
Let us suppose one end of the needle and one end of the pivot to be marked. Let a line, real or imaginary, be drawn on the needle, which we shall call the Line of Collimation. The position of this line is read off on a vertical circle. Let be the angle which this line makes with the radius to zero, which we shall suppose to be horizontal. Let be the angle which the magnetic axis makes with the line of collimation, so that when the needle is in this position the line of collimation is inclined to the horizontal.
