27 
(x, y, 3). And the components of the total force exerted by the mag- 
net on the magnetic element are 
— x) pd b — y) pds ¢ — 8)uds 
nf a a nf ue ; nf oe 
The earth’s radius and that of the luminary being small in comparison 
with their distance, the foregoing expressions are found to be reducible 
to : 
2m cos a -— mf cos B — mM cos ¥ | 
ie? 1 D : 
in which D denotes the distance of the centre of the magnet from the 
centre of the earth, and a, B, y the angles which its axis makes with 
the three axes of co-ordinates, and in which 
Y= fusds, 
the integral being taken between the limits s = + /, J being half the 
length of the acting magnet. 
Now, in place of a single magnet, let there be an indefinite number, 
distributed in any manner throughout the acting magnetic body. Then, 
the radius of this body being small in comparison with its distance, the 
variations of D, both in magnitude and direction, may be neglected, 
and we have, for the three components of the acting forces, 
2mP -mQ —- mk 
Bidens SuecticeeaaeD shat nik ol $e 
in which 
P = =(I cos «), Q = =(1 cos B), R& = (I cos +). 
In order to determine the effects of these forces upon a freely sus- 
pended horizontal magnet, they must be resolved into three others,— 
two of them in the plane which touches the earth at the point m (one 
in the meridian, and the other perpendicular to it), and the third in the 
direction of the earth’s radius. The moment of the two former to 
turn the needle is equal to the moment of the earth’s force by which 
it is opposed, or by mUA6é sin 1’, in which Uis the horizontal compo- 
nent of the earth’s force, and 6 the magnetic declination. We thus ob- 
tain an expression of the form, 
oa 
DST 
in which A, B, and C are known functions of P, Q, R, and ofthe 
latitude and magnetic declination at the place of observation. Simi- 
lar results are found for the changes of the two components of the ter- 
restrial magnetic force. 
Aé 
(4+ B sin 6+ C cos 4); 
