MAGNETIC OBSERVATIONS. 125 



From (11) we easily get 



sin 5 =31 cos 8 + 23 sin f ' + g cos £' + J) sin (£ -f £') -f g cos (£+£') (12) 

 = 21 cos 8 + 33 sin r+ 6 cos £"-j- 3D sin (2£'-f- 8) -J- (g cos (2£'-f- 5) 



Of the last three equations (11) is used when the deviations are given on the 

 correct magnetic points, (12) when the deviations are given on the compass points 

 affected by deviation. 



Equation (12) may be put under the following form, which is sometimes con- 

 venient, and which is very nearly exact, viz. : 



sina = 1 _ I) 1 cos9 / |2(+gsin^-f-gcos^ + T) S in2^ + (vcos2^1 (12a) 



By means of the expressions for sin 8 we may calculate the values of the coeffi- 

 cients 21, 23, (£, £), (5, if we know the deviations on five points. If we have the 

 deviations on more than five points, we may determine the most probable values 

 of the coefficients by the method of least squares; but the calculation will in 

 general be long and difficult. 



If, however, the compass points on which the deviations are given divide the 

 circumference into equal parts, we may determine the exact coefficients 21, 58, £,T>, G, 

 with great ease, and a sufficient degree of approximation, by determining first the 

 approximate coefficients A, B, C, D, E, and then deducing from them the values 

 of the exact coefficients. For that purpose we proceed as follows: 



If the coefficients are less than 20° their squares and products may be neglected, 

 and equation (12) may be put under the form 



S = A + BsinZ'+CcosZ;'+D sin 2£'-\-E cos 2£" (13) 



Let \ hi <5 2 . . . 5 31 be the deviations observed on the 32 points, by compass, $ S 2 S 3 



... S, the natural sines of the rhumbs or of the angles 11° 15', 22° 30' 78° 45' 



respectively, then if the observations have been made on the 32 points we have 

 the following 32 equations from which to determine A, B, C, D, E. 



