Terrestrial Magnetism. 161 
That used by Coutomp was of uniform and very small dimensions throughout its en- 
tire length, and in the calculation was treated as a mathematical line. ‘There are 
other difficulties connected with the practice of this method, on which it is unnecessary 
to dwell: so that, notwithstanding its high sanction, it has not, as far as [ am aware, 
been adopted by any other observer. 
I have alluded to the method of Coutoms in this place, because it suggested that 
which I am now about to propose. It occurred to me that the components of the 
magnetic intensity might both be determined statically, and by one and the same pro- 
cess. Thus all the objections, to which Coutoms’s method is liable, would be avoided, 
and the process itself rendered much simpler of application, as well as more accurate 
in its results. 
Let us conceive a magnetic needle suspended as the ordinary dipping needle, and 
placed in the magnetic meridian ; and let us calculate the directive effect of the ter- 
restrial magnetic force upon it in any position. 
If ¢ denote the quantity of the magnetic fluid in any point of the needle, and » 
the terrestrial magnetic force acting on the unit of quantity, ¢ q will be the force ex- 
erted on the point in question. Let é denote the angle which the direction of this 
force makes with the horizon, or the dip, and let @ be the angle formed by the needle 
itself with the horizon : then é—@ is the angle which the direction of the force makes 
with the needle ; and consequently, the moment of the force tending to turn the 
needle round its axle is ¢ q7 sin (©—@); r being the distance of the point in question 
from the axle. Now, to obtain the total effect of these forces acting on all the points 
of the needle, we have only to multiply this moment by the element of the mass, dm, 
and integrate the result. The total directive effect of the earth upon the needle is 
therefore 
S ¢ qrdm. sin (8—6) = ¢sin (6-0) f grdm. 
The integral /grdm, in this expression, will depend on the form of the needle, and 
on the law of the magnetic distribution in it. Let its value, taken within the limits 
of the dimensions of the needle, be denoted by «: then the moment of the terrestrial 
magnetic force, tending to turn the needle round its axle, is expressed by 
po sin (6-86). 
Now, if a weight be placed on the southern arm of the needle, its effect to turn the 
needle in the opposite way is 
pm Cos 8; 
n being the moment of the mass, or the weight multiplied by its distance from the 
axle. Accordingly, in the case of equilibrium, we have » cos 0 = ¢o sin (¢—8) 
Or, u = ¢o (sin é—cos é tan 6) (1) 
