76 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
made by radiants drawn from two given points (in the present case the magnetic 
poles) with the line joining those two points : and it was suggested to my mind seve- 
ral years ago when considering a question proposed by Professor Wallace in Pro- 
fessor Leybourn’s Mathematical Repository, viz., to “ rectify the magnetic curve.” 
The modification I have here used consists in the expression of the differential coeffi- 
cients of a rectangular equation in terms of the polar angles, 0, and 0 n and the con- 
stants of the given equation : but I hope soon to complete a dissertation on coordi- 
nation generally, and to give the necessary differential expressions that are requisite 
in the investigation of loci, plane, spherical, and solid ; in which case several of the 
following processes may be considerably abbreviated. 
I was compelled to employ the method here specified in consequence of the com- 
plicated form under which the rectangular and polar equations of the magnetic curve 
present themselves, being such as not to encourage the least hope of effecting my 
object by means of either of them, or by both of them conjointly. 
From these investigations it appears, That both systems of branches, the convergent 
and the divergent, are comprised in the same angular equation of the magnetic curve 
already referred to, and deduced at page 238 of the Philosophical Transactions of last 
year : that the divergent branches on one side of the magnetic axis are continuous 
(algebraically and geometrically) of the convergent branches on the other, to the same 
parameter (3 : that the divergent branches are asymptotic, and the geometrical con- 
struction of the asymptote is very easy : that the continuous branches have the poles 
for points of inflexion, and that these are the only points of inflexion within finite 
limits, of the whole system : that the geometrical construction of a tangent at any 
point, that is the direction of a small needle whose centre is at that point, is always 
possible, and the process very simple : together with other properties not less inter- 
esting, though less easy to express in brief phraseology. An elegant curve is thus 
brought within the domain of geometry, which, when its properties are fully deve- 
loped, will, I think, be second only to the conic sections themselves in point of ma- 
thematical interest : whilst its adaptation to at least one important physical inquiry 
will tend to enhance its value in the estimation of those who take an interest in such 
applications as are now, or may hereafter be, made of it, and even in this respect 
render it not inferior in point of value to any other loci except the conic sections, 
and perhaps the logarithmic curve*. 
As both systems of branches of the magnetic curve are found to be involved in the 
same angular equation, and in the same rectangular one also by means of the double 
r It is certainly a remarkable circumstance, that so few of our most elegant curves (geometrically considered) 
are capable of being rendered subservient to physical inquiries : for with the exceptions above mentioned, there 
is, besides the cycloid and the harmonic curve, with perhaps one or two of the spirals, scarcely one which could 
not be expunged from our geometry without any serious injury to physical science. This, together with the 
fact that in the dynamical problems which occur in physics, it is found to be generally most convenient to 
assume the time as the independent variable, has led some writers, too hastily as it appears to me, to conclude 
