XX 



APPENDIX. 



1. Smith's Deduction of Practical Formulae from Poissoris Mathematical 



Theory. 



Let the components of the terrestrial magnetic force*, parallel to three 

 rectangular lines of reference fixed with reference to the ship, be denoted 

 by X, T, Z ; the components at the point occupied by the compass f of 

 combined magnetic force of earth and ship by X', T', Z' ; the components 

 of that part of the ship's action depending on " permanent " or " subper- 

 inanent " magnetism, by P, Q, R, quantities which mathematically must 

 be regarded as slowly varying parameters, their variations to be deter- 

 mined for each ship by observation ; and the components of that part of 

 the ship's action which depends on transiently induced, magnetism by 

 p, q, r so that we have 



X'=X+j? + P, Y'=Y+? + Q, Z'=Z+r+B (1) 



Lastly, let (p, ai) 9 (q, #), (r, a?) be the values which p, q, r would have if 

 the earth's force were of unit intensity, and, in the direction of x ; 



(PI y\ (& y\ (>% y) tne same for y ; an( i (p, *)> fe *) ( r > z ) tne same for * 



By the elementary law of superposition of magnetic inductions the actual 

 value of p will be (p, an) X -f- (p., y) Y -f- (p, z) Z ; and corresponding ex- 

 pressions will give q and r. Hence, and by (1), we have 



X' = X + O,*)X + (p,y)T + (p,z)Z + P, ^1 



Y' = Y + (< ? ,*)X+( g r,y)Y+ fey)Z + Q,l .... (2) 



Z' = Z + (r, a;)X + (r, y} Y + (r, *)% + E.J 



These equations were first given by Poisson in 1824, in the fifth volume 

 of the Memoirs of the French Institute, p. 533. From these Smith 

 worked out practical formulae for the main case of application, that of a 

 ship on even keel, thus : let 



H be the earth's horizontal force ; 



H' the resultant of the earth's and ship's horizontal forces ; 



Q the dip ; 



the ship's " magnetic course ; " 



' the ship's " compass course ; " 



3 = ' the deviation of the compass. 



* That is to say, the force experienced by a unit magnetic pole. The direction of the 

 force is taken as that of the force experienced by a south pole, or (according to Gilbert's 

 original nomenclature) the pole of a magnet which is repelled from the southern regions 

 of the earth. British instrument-makers unhappily mark the north pole with S and 

 the south with N. 



f The length of the needle is supposed infinitely small in comparison with the distance 

 of the nearest iron of the ship from the centre of the compass. 



