Compass produced by the rolling of the Ship. 367 



and 







g-i horizontal. 



Hence 



h. . . 1 

 k= j i, approximate] y. 



Hence, when the ship heads north or south, the amount of the 

 kinetic- equilibrium error is approximately 



Z h. 



11/ U 



Suppose, for example, the period of the rolling to be 6 seconds* 

 (or three times the period of the " seconds' pendulum ") ; I will 

 be 29 feet (or nine times the length of the seconds' pendulum). 

 And suppose the compass to be 14J feet above the axis of roll- 

 ing. We have K—\i (so that the range of apparent rolling in- 

 dicated by a pendulum hung from a point in the position of the 

 bearing-point of the compass is greater by half than the true 

 range of the roll). On these suppositions the kinetic-equilibrium 

 error amounts to 



1 Z . 



2H*' 



About the middle of the British Islands the magnetic dip is 70°, 



and therefore ^ (being the natural tangent of the dip) is equal 



to 2*75 nearly. Hence the kinetic-equilibrium error for the 

 supposed case amounts in this locality to about a degree and 

 three eighths for every degree of roll. 



In an iron ship the equilibrium value of the rolling-error will 

 be approximately the sum of the kinetic error investigated above, 

 and a heeling-error found by an investigation readily worked 

 out from that of Archibald Smith in the f Admiralty Compass 

 Manual' (edit. 1869, Section IV. pages 82-89, and Appendix, 

 pages 139-150), with modification to take into account the de- 

 viation of the apparent level, at the place of the compass, from 

 the true gravitation-level. 



I have used the expression t: kinetic-equilibrium error" to 

 distinguish the error investigated above from that actually exhi- 

 bited by the compass. It is exactly the error which would be 

 shown by an ideal compass with infinitely short period of vibra- 

 tion. A light quick needle (either with silk-fibre suspension, or 



* This would be the case for a ship of any size exposed to regular waves 

 of length 184 feet from crest to crest, and, if moving through the water, 

 moving in a line parallel to the lines of crests. 



