OF MAGNETIC DECLINATION. 355 



3dly, At each observation we may also register the values of the angles 

 7 X and 7 3 , the inclinations of the collimator- wires to those of the theodolite, as 

 found by estimation or otherwise. 



10. The readings 8 X and 8 Z will be affected by changes in declination occurring 

 during the observations ; which, however, may be eliminated with sufficient ac- 

 curacy, by combining each reading with the mean of the preceding and subse- 

 quent ones. At each observation, also, the values of B x or (3 Z will be incorrect, 

 owing to imperfect inversion, the errors being y l and y 3 ; but, as the latter angles 

 may be expected to have sometimes positive and sometimes negative signs, the 

 errors in the individual observations may be eliminated more or less completely, 

 by taking the mean of a sufficient number of readings. 



We shall thus obtain the angles 



( p i + (p 3 = 8j_ - <5 3 , ^ and ^ 3 ; and with more or less accuracy, 

 8 t = 6, and 8 Z = 180° + (3. 



Similarly, if the magnet be repeatedly inverted, with the lines 2 and 4 alter- 

 nately coinciding with the line 0, we shall obtain, in like manner, 



02 + 04 = ^2 - $i > ^2 aIld ^4 • 



6 2 = 90° + 8 nearly, and /3 4 = 270° + 8 nearly. 



If now, as before, we suppose the magnetic and optical axes, and a vertical 

 line through the point of suspension, to meet a spherical surface in the points 

 A, B, Z, we shall have, in fig. 7, A l B x Z, A 2 B 2 Z, A 3 B 3 Z, A 4 B 4 Z, for the posi- 

 tions of those points in the four positions of the magnet, and we shall obtain 



sin 4 sin a sin 3 sin a . 



~~ • — a — """■ » ~~ ■ /T - — — • <BC. 



sin p 1 sin 4> x sin p 3 sin «\> z 



from which, substituting the values of 8 V 6 2 , &c, in terms of (3, we have 



sin X sin a sin 3 sin a 



sin (3 sin ^ sin (3 sin ^ 3 ' 



sin <p 2 sin a sin 4 sin a 



Whence 



Therefore 



cos (3 sin 4/ 2 cos 8 sin ^ 4 



sin 8 = S j" 0i + sip sin ^ sin ^ 

 sin "vj/j + sin ^ 3 ' sin a 



cos 8 = sin 02 + sin 04 sin ^ 2 sin 4 4 

 sin ^ 2 + sin 4- 4 " sin a 



tan /3 _ sin | (0 1 + 3 ) cos i (tfc - <ft 8 ) sin fr (^ + »|/ 4 ) cos & (^ 2 - + t ) sin ^ sin ^ 

 sin i (0, + 4 ) cos i (0 2 - 4 ) sin \ (^ + ■+„) cos \ (^ - -vj, 3 ) sin ^ sin ^ 4 



and sin a = sin H0i + 8 )cqb&(0i - 3 ) sin ^ sin ^ 3 



sin J (^ + 4> 3 ) cos I (^ - 4» 8 ) sin 8 



