356 MR WILLIAM SWAN ON OBSERVATIONS 
11. In these formule the angles, in terms of which a and @ are found, are all 
given by observation except ~,—p, and p,—¢,. As, however, these angles are 
small, their cosines will differ little from unity: and since also y,, ,, &c., are all 
nearly equal to 90°, we may evidently, for an approximation, assume all the fac- 
tors on the right hand, except sin} (p,+,) and sin 3(p,+,) as equal to unity. 
Then since 
9; +p, =0,-— 95 and ps + > =9,— 9,, 
we shall obtain ili pcre 
tent ane 6, _ 8.) ‘ 
sin} (6, = 83) or (sma = sin $ (6, — 8). 
‘ste 
ts sin 6 cos ( 
Formule for calculating the Errors occasioned by Imperfect Inversion of a Declinometer 
Magnet. 
12. Having thus found approximate values of a and @, 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, ¢, and @, and hence e, the correc- 
tion to be applied to the observations of the imperfectly inverted magnet. If, 
however, formule for calculating ¢ directly be preferred, the following may be 
used :— 
1. In the erect and inverted positions of the magnet, we have the equations 
sng, _ sin, sing, _ sin, 
sina sin y, ’ sin & sin, ’ 
where ¥y, ¥, are found by observation, also 6,=(6+ +, and §,=6+4,; 
y, and , being found by observation, and a and @ calculated by the formule of 
Art. 11. 
Then remembering that 
?, + >, = 9-4, 
sin @ 
8sin y, sin p, cos 3 (0, — 0,) 
and putting 
cos 0 = 
it may be shown that 
sin € = cos (8 + 4, + (,) — cos(O — y, — B,)+ cos(O — , + B;) — eos (8 + 4, — Bs) 
—cos(O + , + 6,) + cos(@ — Y, — 2.) + cos(O + Y, — B,) — cos (8 — ¥, + B,) 
13. More convenient formule, 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 @ and separately. 
lst, If errors occur in the values of 6 alone, we may suppose y to have its 
correct value, which, as shown in Art. 5, is 90°. 
The equations of last article then become 
sin fd, = sina sin@,, sind, =sina sin @, 
