350 : MR WILLIAM SWAN ON OBSERVATIONS 
porting it rests, or ZO, the torsion-fibre by which it is suspended. We may 
then suppose the magnetic axis to be transferred parallel to itself, until it passes 
through O, without altering the direction which the freely suspended magnet 
will assume; and, in like manner, the observed direction of the magnet will 
remain unchanged, although we suppose the optical axis of the collimator at- 
tached to it to be transferred parallel to itself, until it passes through O. Let 
AA’ represent the direction of the magnetic, and BB’ that of the optical axis, and 
let a spherical surface described about O meet the lines OA, OB, OZ, in the points 
A, B, Z; then the circle Z N z will be the plane of the magnetic meridian, and the 
spherical angle AZB, or the are NC of the horizontal circle NESW, will measure 
the horizontal angle between the optical axis of the collimator and that plane. 
It is obvious, also, that we may include the cases of the ordinary compass-needle 
or of the magnet with an attached mirror, if in the one case, we conceive OB to be 
the axis of figure of the needle ; or ifin the other, we suppose that a pencil of rays 
proceeding from the fixed scale along a given line, such as NO, and falling on the 
mirror at O, is reflected to B. 
If now, when the verniers of the theodolite employed to observe the magnet 
indicate zero, the optical axis of the theodolite-telescope is in a plane parallel to 
the plane ZOD, the arc DN will be the true, and DC the apparent reading for 
the magnetic meridian. 
Hence, if the arcs DN, DC, and NC, be represented by ° 6, and ?. we shall 
have the apparent reading for the magnetic meridian, 
}, =0 aH op. 
Next, let z, instead of Z, be taken as the point of suspension, the angles AOB, 
AOZ, BOZ, all remaining unchanged, then the magnet will be in precisely the 
position which it would occupy if the whole figure revolved about the line NS, 
until z coincided with Z. 
If then a O, 6 O represent the new positions of AO, BO, we shall have 
aZLb=azb= AZB= ®; 
and if eDE[os: 
6, = 6 — 9, 
where 0, is the apparent reading for the magnetic meridian in the new position 
of the magnet. 
Finally, adding, ¢ is eliminated, and we have 
d= £0, mid 65) 5 
which shows, that if the magnet be inverted, in the manner described, the mean 
of the readings in the erect and inverted positions will give a rigidly accurate 
result. 
