104 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
With centre O, and distance O A, describe the circle A a A', and with the ratio 
A T : A IJ, the circle A A 2 Aj ; and, as in the former cases, these arcs will be finite. 
Join A ( O, cutting the magnetic axis in K r Then also none of the needles either in 
A t A., A or in A G, except that at A, will pass through O, and that at A ; passes 
through I\. 
In the same way, by producing OA ( to B till BT:BU is the triplicate of the ratio 
K, T : K ( U, we shall have another point B in the curve of verticity ; through which, 
with centre O and distance OB, describe a circle cutting the circle of ratios of B in 
the points B and B ; ; and join B ( O, cutting the magnetic axis in K 2 . And repeat 
this process as far as may be necessary both as to construction and reasoning. 
Then, since the distances TS, S K, K I\ l5 1^ K 2 , .... are all finite, and the distance 
T R also finite, a continued repetition of these processes will at length conduct us to a 
point K n , either coincident with R, or more remote from T than R is. Let E be a 
corresponding point in the curve of verticity. Then the segment of the curve joining 
N and A must lie in the mixtilineal angle formed by the line A N' and the arc N'Q 2 N ; 
the segment A B is in the mixtilineal angle B A, A, and so on to E. But the arc of 
the magnetic meridian lies wholly without this series of angles, and hence cannot in 
any one point coincide with the segments of the curve which lies within them. The 
magnetic meridian, therefore, can only cut the asymptotic branch of the curve which 
lies to the left of T and above the axis, in one single point N. 
In the same way, exactly, may it be shown that the magnetic meridian can only 
cut the other asymptotic branch to the right of U in one single point. 
By processes of the same nature it may be proved that the finite branches of the 
convergent system can never be cut in more than two points by a circle whose centre 
is () ; and that the same is true to the divergent system. But neither of these cases 
falls within the objects of the physical problem under consideration, it would be su- 
perfluous to enter upon them here; although for giving completeness to the geo me- 
trical problem such a discussion would be indispensable. However, after what has 
been done in the foregoing pages, this portion of it can present no difficulty to the 
geometer who may be disposed to follow it out, as the reasonings which I have em- 
ployed in its solution, and which completely apply to all the cases, is essentially the 
same as that detailed in the case here discussed at length. It is only necessary to 
observe, that the positions of the finite and infinite branches in the two systems are 
so situated that, in the divergent system, all four branches, the two finite and the two 
infinite ones, may be cut by the same circle, or only the two infinite ones, depending 
upon the radius of that circle : whilst in the convergent system, which we have had 
occasion here to consider, only two of the branches, either the two infinite ones, or 
the two finite ones, or one of each, can be cut by the same circle. 
XXXII— Tl ic consequence of all this investigation, then, is, that if we admit the 
hypothesis of two centres of magnetic force situated within the earth, there will be two, 
