VI 



1. Harmonic reduction of observations. 



2. Practical expression of the full mathematical theory. 



3. Heeling error. 



4. Dygograms. 



5. Eule for positions of needles on compass card, with dynamical and 

 magnetic reasons. 



1. Harmonic reduction of observations. The disturbance of the compass 

 produced by the magnetism of a ship is found by observation to be the 

 same, to a very close degree of approximation, when the ship's head is 

 again and again brought to the same bearing, no great interval of time 

 having intervened, and no extraordinary disturbance by heavy sea or 

 otherwise having been experienced in the interval. Overlooking these 

 restrictions for the present, we may therefore say, in Fourier's language, 

 that the disturbance of the compass is a periodic function of the angle 

 between the vertical plane of any line fixed relatively to the ship, and 

 any fixed vertical plane, when the ship, on " even keel " or with any con- 

 stant inclination, is turned into different azimuths the period of this 

 function being four right angles. Hence also the disturbance of the 

 compass is a periodic function of the angle between the vertical plane of 

 the chosen line moving with the ship, and the vertical plane through 

 the magnetic axis of the compass. The line moving with the ship being 

 taken as a longitudinal line drawn horizontally from the stern towards 

 the bow, and the fixed vertical plane being taken as the magnetic meri- 

 dian, the angle first mentioned is called for brevity " the ship's magnetic 

 course" and the other " the ship's compass course." 



One of Smith's earliest contributions to the compass problem was the 

 application of Fourier's grand and fertile theory of the expansion of a 

 periodic function in series of sines and cosines of the argument and its 

 multiples, now commonly called the harmonic analysis of a periodic 

 function. To facilitate the practical working out of this analysis, he gave 

 tables of the products of the multiplication of the sines of the " rhumbs " 

 by numbers, and by arcs in degrees and minutes ; also tabular forms and 

 simple practical rules for performing the requisite arithmetical opera- 

 tions. These tables, tabular forms, and rules, just as Smith gave them 

 about thirty years ago, are in use in the Compass Department of the 

 Admiralty up to the present time. From every ship in Her Majesty's 

 Navy, in whatever part of the world, a table of observed deviations of the 

 compass, at least once a year is sent to the Admiralty, and is there sub- 

 jected to the harmonic analysis. The observations having been accu- 

 rately and faithfully made, a full history of the magnetic condition of the 

 ship is thus obtained, and want of accuracy, or want of faithfulness, if 

 there has been any, is surely detected. The rigorous carrying out of this 

 system, with all the method and business-like regularity characteristic 

 of the scientific departments of our Admiralty, has undoubtedly done 



