To bring the compass needle back to North it would be necessary to move up nearer the compass dial the fore-and-aft magnet (shown below), whose magnetism would act on the compass needle on this heading of the ship exactly as the athwartship magnet acted on the compass needle when the ship was headed North:
Now the ship's compass has been corrected for the North and East headings respectively. The next correction is for the heading half way between, i.e., North-east. If there is any Deviation on this heading, adjust the cast iron cylinders (called Quadrantal Correctors), which are on each side of the compass bowl, by moving them toward or away from the compass until the ship's head by compass is North-east at the proper time and bearing by pelorus.
The ship's compass has now been corrected for one whole quadrant, namely, from North to East, and this will suffice for all four quadrants since the relationships of the magnets themselves and the magnetism of the compass needle is the same for any of the other three quadrants as for the first. Compass adjustment, however, can never be absolutely accurate. For that reason, it is wise to steam the ship completely around, steadying on every fifteen degrees by pelorus to determine and keep a record of remaining errors.
There is one more correction to make, i.e., for the heeling error. This correction is necessary in case the ship is yawing in a sea-way so much that the relationship of the ship's magnetism to the compass needle is decidedly different from what it is when the ship is on a comparatively even keel. It is compensated by a vertical magnet directly underneath (or over) the binnacle, details in regard to which can be secured from Bowditch Art. 125, p. 53.d from Bowditch Art. 125, p. 53.
It must be borne in mind that compass adjustment is not an exact science, that an adjustment for one latitude is not correct for another, that anyone of a hundred different causes can affect the magnetism of the ship or of the compass needle, which in turn directly affects the Deviation. In this connection, it would be well to read Bowditch Art. 129, p. 55. You should also read Arts. 119-130 in which are given, more fully and in more scientific language, the contents of this lecture.