of the Eurth’s magnetic Force in absolute Measure. 8) 
Now, if p denote the distance between the centres of the two magnets, 
a@ =v +r’ — 2prcos y. 
Wherefore, developing the inverse powers of a@ in series ascending according 
to the powers of = , stopping at the inverse fifth power of the distance, and substi- 
tuting in the expression for the moment above given, it becomes 
4M) 
Dan 
Me sin ¥}u (2 + 90s ¥ ~ + 6(5 cos? — y5)+ 
This being the moment of the force exerted by the deflecting magnet upon a 
single particle, m, of the suspended magnet, the moment of the force exerted upon 
the entire magnet is obtained by multiplying by dr, and integrating between the 
limits 7 = + /, / being half the length of the suspended bar.* The magnetism 
being supposed to be distributed symmetrically on either side of the centre of the 
suspended magnet, and the axis of suspension to pass through that centre, we have 
( mriar == (0: 
+1 41 
Accordingly, denoting the integrals ( mrdr, ) mr’dr, by M’ and M,’, the ex- 
—!l 
pression for the moment of the whole force is 
2mm’. Ms = M;\ 1 
“ sin y {1 +(24 3(5 cosy — 1) *,) =H. 
When there is equilibrium, this moment must be equal to that of the force 
exerted by the earth upon the suspended magnet. Let x be now taken to denote 
the horizontal component of the earth’s magnetic force. The moment of the 
force exerted by that component upon the particle m of the suspended magnet is 
MXP SIN U 5 
u denoting the deviation of the axis of the magnet from the direction of the 
force. Multiplying by dr, therefore, and integrating, the total moment is 
* We have here assumed, that the effect of a magnet is the same, with respect to any point at a 
moderate distance, as if the magnetic elements in each section perpendicular to its axis were all con- 
centred in the axis; or, in other words, that the integration with respect to the other dimensions of 
the bar introduces no new term into the integral. There is no difficulty in proving that such is the 
case, the magnetic elements being supposed to be distributed symmetrically with respect to the axis. 
VOL. XXI. Cc 
