106 MR. DAVIES’S INVESTIGATIONS CONCERNING TERRESTRIAL MAGNETISM. 
the results of the hypotheses of like and unlike poles are comprised in the same equa- 
tion, it will, as was the case in the present discussion, require a peculiar mode of treat- 
ment to separate and define the curves of equal intensity belonging to each of them. 
XXXIV. — It will be remarked, that in the equations of the magnetic curve, the 
final position of the needle, or that which it takes when its centre is coincident with 
either centre of force, is different from that which is exhibited by a needle acted on 
by an artificial magnet in all our experiments. This might at first sight seem to throw 
some doubt on the validity of the principle employed in deriving the equation of the 
magnetic curve ; but a little reflection will convince us that the conditions of such 
experiments are different from those which obtain in the case before us. 
In all our experiments, the length of the needle itself bears a finite ratio to the di- 
stance of its centre from the poles of the magnet upon which we experiment, and 
hence the action becomes mutual the two magnets, and the system of actions being 
thus rendered compound, its results must be expressed by a more complicated for- 
mula than in the case we have supposed, and from which our equations of the mag- 
netic curve have been derived. This complexity of the conditions implies a corre- 
sponding complexity of the equation by which they are expressed ; but uniform ex- 
perience has shown that as we diminish the length of the needle, and increase its 
distance from the poles, the observed results are more nearly approximative to the 
results of the hypothesis from which we have started. In the case of the curves exhibit- 
ed by iron filings strewed on a paper above a bar magnet, the approach of the observed 
curve to the calculated one is very close*. But in this case the length of the needle 
(or magnetized particle of the iron) is very small in comparison with the magnet and 
with its distance from the magnetic poles ; and, moreover, has in itself so little 
magnetic intensity as to exert an insensible reciprocal influence on the state of the 
bar itself. This is precisely a miniature representation of the case of a small needle 
acted on by the terrestrial magnet, though the ratio of the particle of iron to the 
magnetic bar is many times greater than that of a needle to the terrestrial magnet ; 
and hence the discrepancy between the observed and calculated result in the former 
case is many times greater than in the latter. No ground of exception to our plan of 
inquiry can therefore be found in this circumstance, but rather a confirmation of its 
validity in reference to the use we have made of it. 
* Leslie’s Geometrical Analysis, p.405. 
Royal Military Academy, Woolwich, 
January 2 7th, 1836. 
