Compass produced by the rolling of the Ship. 365 



tion in which it would rest under the magnetic forces and a force 

 of apparent gravity equal to the resultant of yWand a force aW 

 in the direction opposite to that of a. Now the weight of the 

 compass is so great and its centre of gravity so low that the level 

 of the card is scarcely affected sensibly by the greatest magnetic 

 couple experienced by the needles'*. Hence in kinetic equili- 

 brium the plane of the compass-card is sensibly perpendicular 

 to the direction of the " apparent gravity " defined above ; and 

 the magnetic axis of the needles is in the direction of the result- 

 ant of the components, in this plane, of the magnetic forces of 

 earth and ship. Hence it is simply through the apparent level, 

 at the place in the ship occupied by the compass, differing from 

 the true gravitation-level, that the problem of the kinetic-equi- 

 librium position of the compass in a rolling ship differs from the 

 problem of the heeling-error referred to above. That we may 

 see the essential peculiarities of our present problem, let there 

 be no magnetic force of the ship herself or cargo. The kinetic- 

 equilibrium position of the magnetic axis of the compass will be 

 simply the line of the component of terrestrial magnetic force in 

 the plane of the apparent level. Let ic be the inclination of this 

 plane to that of the true gravitation-level, and cf> the azimuth 

 (not greater than 90°) from magnetic north of the line LL ; of 

 the intersection of the two planes (a diagram is unnecessary) ; 

 also let H and Z be the horizontal and vertical components of 

 the terrestrial magnetic force. The component of this force in 

 the plane of apparent level will be the resultant of H cos cf> along 

 LL' and H sin <j) cos k + Z sin k perpendicular to LL' ; and there- 

 fore, if (j>! denote the angle at which it is inclined to LL', we have 



, H sin cf> cos k + Z sin k , , Z sin k 



tan d>,= ■h rr ~ — = tan 6 cos k+ -^ r . 



r/ Hcos$ r Hcos<£ 



If, as usual in compass questions, we reckon the directions as of 

 forces on south magnetic poles (or the northern ends of the com- 

 pass-needles), the direction of H cos <f> is along LL' northivards, 

 and the direction of Z sin tc, when the ship is anywhere north of 

 the magnetic equator, is downwards in the plane of the apparent 

 level. 



Now, as we are only considering the effect of rolling, the di- 

 rection of the given "acceleration" of the bearing-point will 

 always be in a plane perpendicular to the ship's length ; and 

 therefore LL' will be parallel to the length. (It will in fact be 

 the line through the " lubber-points" of the compass-bowl.) 

 Hence, and as compass angles are ordinarily read in the plane 

 of the fly-card, the kinetic equilibrium-error of the compass is 



* Generally no adjusting counterpoise for the compass is required when 

 a ship goes from extreme north to extreme south magnetic latitudes. 



