356 ME, WILLIAM SWAN ON OBSERVATIONS 



11. In these formulae the angles, in terms of which a and ft are found, are all 

 given by observation except (p 1 — (f> 3 and cp 2 — <£ 4 . As, however, these angles are 

 small, their cosines will differ little from unity : and since also 4^, 4, 2 , &c, are all 

 nearly equal to 90°, we may evidently, for an approximation, assume all the fac- 

 tors on the right hand, except sin£ (<^> 1 + <p 3 ) and sin %((p 2 + $ 4 ) as equal to unity. 

 Then since 



0i + 03 = $i - & and cj) 2 + 4 = 8 2 - 8 t , 

 we shall obtain 



tan/3 = sin H^i ~3 8 ) , 

 sin \ (8 2 - 8 t ) 



sin h (8, — £,) sin A (8 9 — 8.) 



sin a = a —j or sm a = — z y 2 ^ . 



sin ft cos ft 



Formula? for calculating the Errors occasioned by Imperfect Inversion of a Declinometer 



Magnet. 



12. Having thus found approximate values of a and ft, it will be seen, on 

 referring to the equations of Art. 6, that we are in possession of all the data ne- 

 cessary for calculating, from those equations, (p 1 and (p 3 and hence e, the correc- 

 tion to be applied to the observations of the imperfectly inverted magnet. If, 

 however, formulae for calculating e directly be preferred, the following may be 

 used : — 



1. In the erect and inverted positions of the magnet, we have the equations 



sin <p 1 sin ft x sin (p 3 sin ft 3 



sin a sin 4^ ' sin a sin 4> 8 ' 



where 4^ 4* s are found by observation, also /3 1 = ft + y v and ft z = ft + y 3 ; 

 y x and y z being found by observation, and a and ft calculated by the formulae of 

 Art. 11. 



Then remembering that 



^ + cf> 3 = 8 1 - 8 3 , 

 and putting 



/, sin a 



cos v = 



8 sin 4^ sin 4^ cos ^ (8 l — 8 3 ) 

 it may be shown that 



sin e = cos (0 + 4^ + ft 3 ) - cos (0 — 4^ — ft 3 ) + cos (0 — 4^ + /3 3 ) — cos (0 + 4^ — /3 3 ) 



-COS(0 + 4, 3 + /? 1 ) + COs(0- VJ.3-/3J + COS (0 + 4^3 "A) "COS (0-4/ 8 4-/3,) 



13. More convenient formulae, however, may be obtained by calculating the 

 errors in the resulting value of the magnetic declination on the supposition that 

 variations take place in the values of ft and 4, separately. 



1st, If errors occur in the values of ft alone, we may suppose 4, to have its 

 correct value, which, as shown in Art. 5, is 90°. 



The equations of last article then become 



sin (f) l = sin a sin ft x , sin (p 3 = sin a sin ft 3 



