366 Sir William Thomson on the Perturbations of the 



exactly equal to $, — $> When k is a small fraction of 57 0, 3 

 (the "radian/ 5 as the angle whose arc is equal to radius has 

 been called by Professor James Thomson), which is the case ex- 

 cept in extreme degrees of rolling when the compass is properly 

 placed*, we have approximately 



<£ / — <£==#-- cos (p. 



The direction of this error is, for the northern ends of the needles 

 in the northern magnetic hemisphere or for the southern ends 

 in the southern hemisphere, towards the side on which the appa- 

 rent level is depressed — that is (as practically the compass is 

 always above the axis of rolling), towards the elevated side of the 

 ship. It has its maximum value 



Z 



*H 



when c5 = ; that is to say, when the ship heads north or south 

 magnetic. To estimate its amount, consider perfectly regular 

 rolling ; which in general fulfils approximately the simple har- 

 monic law, so that we may put 



i = Isinn^ 



where i denotes the inclination of the ship at time t 3 and n and 

 I constants. Let h denote the height of the bearing-point of the 

 compass, vertically above the axis of rolling when the ship is ver- 

 tical. For the amount of its acceleration we have 



or 



Now, if / denote the length of a simple pendulum isochronous 

 with the rolling of the ship, we have 



H> 



and therefore 



h. 

 1 



■9l •«• 



The direction of a, being tangential to the circle described by 

 the bearing-point, is approximately horizontal ; and therefore the 

 direction of apparent gravity will be approximately that of the 

 resultant of 



g vertical, 



* The " mast-head compass/ 5 perniciously used in too many merchant 

 steamers, may, in moderate enough rolling, experience deviations of appa- 

 rent level amounting to 20° or 30° on each side of the true gravitation-level. 



