420 Royal Society. 
he finds, 1°, that the two equations, the convergent and the divergent, 
or that in which the poles are unlike, and that in which they are 
like, are both expressed by this equation, and essentially included 
in it: 2°, that the divergent branches on one side of the magnetic 
axis are algebraically and geometrically continuous with the con- 
vergent branches on the other side; the parameter (6) being the 
same in both cases: 3°, that the divergent branches are assym- 
ptotic, and the assymptote is capable of a very simple construction ; 
4°, that the continuous branches have the poles as points of in- 
flexion, and that these are the only points of inflexion within finite 
limits: 5°, that a tangent at any point of the curve, or, which is 
the same thing, the direction taken by a small needle placed there, 
admits of easy construction: 6°, that when the parameter (() is 
such as to cause the convergent and divergent branches to intersect, 
they do so in a perpendicular to the magnetic axis drawn from the 
poles: 7°, that the convergent branches are always concave, and 
the divergent always convex, to a line at right angles to the magnet, 
drawn from its middle,—besides other properties not less interesting, 
though less capable of succinct enunciation. 
Having separated the branches belonging to the case of like poles 
from those belonging to the unlike ones in the magnetic curve, the 
author proceeds to asimilar separation of the corresponding branches 
in the curve of verticity. In the former case the curve is composed 
of two branches infinite in length, having the magnetic axis for as- 
symptotes, lying above that axis, and emanating from the poles to 
the right and left ; and of two finite branches, continuous with those 
just described, and lying below the magnetic axis; one of which 
passes through the centre of the earth, and meets the other in the 
perpendicular from the middle of the axis; so that the whole system 
is constituted by one continuous curve, extending from negative 
infinite to positive infinite, and having the lines drawn from the centre 
of the earth to the magnetic poles as tangents at the poles ; and no 
part of the curve lies between these tangents. It bears in form 
some general resemblance to a distorted conchoid ; this curve not 
having either cusp or loop. In the second case, the curve is also 
composed. of four branches, two finite and two infinite ones; the 
latter having the line drawn from the centre of the earth through 
the middle of the magnet as assymptotes, and both lying on the 
same: side of it as the more distant pole; and the finite branches 
joining these continuously at the poles, and each other in the mid- 
dle of the magnetic axis; the one from the nearer pole lying above 
the axis, and the one from the remoter pole lying below it. The 
branches, where they unite at the poles, have the lines drawn from 
the centre of the earth to the poles as tangents, and the Jower in- 
finite branch passes through the centre. The whole system of 
branches is comprised between the polar tangents; and the two 
systems are mutually tangential at the poles, and intersect each 
other at the centre; but they have no other point in common. 
Lastly, the author proceeds to demonstrate that a circle (namely, 
the magnetic meridian) described from the centre of the curve of 
verticity, will always cut the convergent system in two points, but 
