376 Sir. AV. J. LIacquorn Raxkine on the Magnetic Meridian. 



Note on the Determination of the Magnetic Meridian at a Distance from 

 Land. By W. J. Macqtjokk Rankixe, C.E., F.R.SS. L. & E, 



The principle of the following method of approximately finding the 

 magaetic meridian on board ship when at a distance from land, and some 

 of the results of that principle, were communicated to the Royal Society 

 in 1853, and an abstract of them published in the Proceedings of that 

 body. In the present note the theoretical investigation is somewhat 

 simplified, and the resulting formula is put into a shape more convenient 

 for practical use ; and a method is also explained of substituting a geo- 

 metrical construction for some of the calculations. 



The principle in question is this: — When a ship performs a complete 

 rotation, her head returning exactly to the point from which it started, the 

 sum of the mechanical icork performed by the horizontal rotative forces 

 acting between the compass-needle and the earth, betiveen the earth and the 

 ship, and hetioeen the needle and the ship respectively, is equal to nothing. 



Let the following symbols denote angles, measured in the direction of 

 motion of the bands of a watch, i.e., from north to east: — 

 S, the angle from the north end of the needle to the magnetic north = 



the westerly deviation of the needle, when positive. 

 ^, from the magnetic north to the ship's head = true magnetic bearing of 



the ship's head. 

 ^' = p-\-l, from the north end of the needle to the ship's head = apparent 

 magnetic bearing of the ship's head. 

 Let the horizontal couples, or rotative moments, tending to vary the 

 above angles, be denoted as follows : — 

 P, the couple acting between the needle and the earth 



(= m X sin o, where m is the magnetic moment of the needle, and 

 X the earth's horizontal force) ; 

 Z, the couple acting between the earth and the ship ; 

 P, that acting between the needle and the ship (being equal and op- 

 posite to that acting between the needle and the earth). 

 Then the principle stated above is thus expressed : — 



-fvdi-f;'zd^+fl-'Fdc = o (L) 



Now the first term of this equation,/*P d S, being obviously null, it is 

 reduced to 



/-ZdC=/fPdC = mX/rsinS-d^- 



or, dividing both sides by 2 a- m X, 



A denoting, as in Mr. Archibald Smith's notation, the mean of the sines 



