352 MR WILLIAM SWAN ON OBSERVATIONS 
sin(A +1) _ Sinn , 
Hence sin (A’ +s) Sina 
Now if the inversion has been accurate, according to definition, we should have 
A= 180° — A; and © 9 = 180° — 7. 
Therefore 
sin (A + 4) _ 1 
sin (\’ — 1) Tie 
a condition which, as ‘may have all values from 0° to 90°, can only be satisfied 
by A = 90° or A = 270°; that is, when the magnet is suspended with its magnetic 
axis horizontal. Such is the position which the magnetic axis is made to assume 
in practice, with more or less accuracy; and it therefore follows, that the magnet 
may remain in equilibrium after accurate inversion. 
Inaccurate Inversion of a Magnet. 
6. We have next to inquire what will be the errors occasioned by inaccurate 
inversion. 
In fig. 1,let AB=a, BAZ=6, BLZ=%, AZB = $,; 
and after inversion, let 
baZ =f,, bZ = 3, aZLb=9;; 
while ab = AB= 4, 
Also, as before, let 
DN=6, DC=d,, De=6, 
Then the observed angles for the magnetic meridian are 
6,=0+ >, 6,=5— 
0 
From which 
z (6, + 65) 9 z (Pi % fs) 
4 (0, + Ds) = 83 
where the error committed by taking the mean of 6, and 0, for 4, is 
€=3(f — Ps) 
Also in the triangles ABZ, a6 Z, we have 
: _ sina sin sin @ sin (, | 
Ste sin yy, j sin , ; 
from which equations, supposing the other angles to be known, ¢,; and ?,, may 
be calculated, and ¢, the correction to be applied to 6, may be obtained. 
sin Ps; = 
Method of ascertaining the relative positions of the Magnetic and Optical Awes in a 
Oollimating Magnet. 
7. It thus appears that we can calculate the error in the determination of 
magnetic declination due to imperfect inversion, provided we know the angles 
a, 8, and , both in the erect and inverted positions of the magnet. 
