Mr. W. J. SIacquokn Rankine on the Magnetic Meridian. 377 



of all the deviations of the compass-needle (positive when westerly) 

 observed in swinging the ship. 



Now, it is probable that the mean value of Z, the rotative force acting 

 between the earth and the ship, varies simply in proportion to X, the 

 earth's horizontal force ; and, consequently, that A is a constant for a 

 given ship, for all positions on the earth's surface, which may be ascer- 

 tained once for all, when the ship is in port. 



The symbol -^ fl" sin S d f ' denotes the taking of the mean of the sines 



of the deviations of the needle with the ship's head on an infinite number 

 of equidistant apparent bearings. In practice, only a finite number of 

 such deviations can be observed. Let the mean of their sines be denoted 

 by 



tn sin. S = A (3.) 



This quantity having been ascertained, once for all, let the ship be 

 supposed to be at a distance from land, and let it be required to find the 

 magnetic meridian. Let the ship be swung round, and let the apparent 

 magnetic bearings of any fixed distant object be taken, with the ship's 

 head on each of the thirty- two points, or on sixteen equidistant points, 

 as the case may be. A star will answer for an object, if its apparent 

 motion be allowed for by calculation. It is required to find the true 

 magnetic bearing of this object. 



Let a. denote this true magnetic bearing, a.' any one of the apparent 

 magnetic bearings, then the corresponding deviation of the needle is 



and consequently 



sin. 8 ^ COS. a, sin. «' — sin. « . cos. «' 



Therefore, let S denote the mean of the sines of the apparent bearings 



of the distant object, and C the mean of their cosines; then 



A = S • cos. « — C • sin. « (4.) 



the solution of which equation gives, for the ti^ue magnetic bearing of the 



object, 



S . A 

 « = arc. tan. -— arc. sm. ^ ^^_^^2 (5.) 



The true magnetic bearing of a visible object having been thus deter- 

 mined, the magnetic meridian is known, and also the deviations of the 

 needle for all those positions of the ship's head at which apparent bear- 

 ings of the object were taken. 



It is scarcely necessary to add, that in taking the means of the sines 

 and cosines, sines of easterly bearings and cosines of northerly bearings 

 are to be considered as positive ; and sines of westerly bearings and 

 cosines of southerly bearings as negative. 



