368 Sir William Thomson on Perturbations of the Compass. 



supported on a point in the ordinary way) having a period of not 

 more than about two seconds, shows the rolling-error very beau- 

 tifully, taking at every instant almost exactly the position of 

 kinetic equilibrium. I have thus found the rolling and pitching 

 errors so great in a small wooden sailing-vessel that it became 

 very difficult to make exact observations with the quick compass, 

 either in the Frith of Clyde or out at sea on the Atlantic unless 

 when the sea was exceptionally smooth. The well-known kinetic 

 theory of " forced oscillations " is readily applied to calculate, 

 whether for a wooden or an iron ship, the actual "rolling-error" 

 of the compass, from the "kinetic-equilibrium error" investigated 

 above. Thus let 



u be the deviation of the compass at any instant, from the 

 position it would have if the ship were at rest and upright; 



T the period of its natural oscillation if unresisted by any 

 " viscous " influence (the damping effect of copper, intro- 

 duced by Snow Harris and used with good effect in the Ad- 

 miralty standard compass, being included in this category) ; 



2/ a coefficient measuring the amount of viscous resistance ; 



E the extreme equilibrium value of the rolling-error ; 



T' the period of the rolling. 



2ir 2tt 



For brevity put n— -=^ % and n'— j^-. The differential equa- 

 tion of the motion is 



The integral of this proper to express the effect of regular rolling 

 is 



~ ri 2 cos (n't + e) 



U ~ \/{ (n' 8 -»«) 8 + 4nf 2 / a } ' 

 where 



€ = tan~ 1 a % 



•It would extend the present communication too far to enter 

 on details of this solution. For the present it is enough to say 

 that no admissible degree of viscous resistance can make the 

 rolling-error small enough for practical convenience, unless also 

 the period of the compass is longer than that of any consider- 

 able rolling to which the ship may be subjected. Probably a 

 period of from 15 to 30 seconds (such as an ordinary compass 

 has) may be found necessary for general use at sea ; and it be- 

 comes an important practical question how is this best to be 



