CONCERNING TERRESTRIAL MAGNETISM. 
95 
All the properties of the magnetic curve that are essential to our future investiga- 
tions of the physical problem, and none else, have now been fully stated and esta- 
blished. We shall proceed now to their application. 
XXVII. — On the Curve of Magnetic Verticity : or that in any Point of which a Mag- 
netic Needle being placed, it will he directed towards a given Point. 
At each point of the curve of verticity, the tangent to the magnetic curve through 
that point is directed to the given point, this being the defining character of the curve 
of verticity. Its polar equations are given in (81.) and (82.), and before we proceed 
to employ the properties of the magnetic curve to the determination of the properties 
of this, it is necessary to make one or two remarks on those equations, and deduc- 
tions from them. 
In the first place, the general equation (79.) involves the fourth power of r, and 
therefore, generally indicates that a line drawn through any other point in the plane 
of the curve will have four values, either all real, two real, or all imaginary, since a 
transformation of coordinates does not alter its dimension, and therefore the number 
of its roots ; and it is easy to see that in its general form, the separation of its roots 
would be impracticable, in the literal state of the component data. But as by taking 
the coordinates in the particular way that is there done, a loss of two dimensions has 
occurred, or, more properly speaking, a separation of the general equation into two 
others, the utmost simplification that can possibly arise from the mode of assuming 
the system of reference, has been here effected. It is very probable that this is the 
only way in which that separation could have been made ; and hence there is little 
hope of further improvement in the process by the transformation of coordinates, at 
least so far as origin of r is concerned*. 
A 
The point O (Plate XIV. fig. 14.) is a quadruple point, whilst any line drawn through 
O can cut the curve in only two points besides O. The values of r can therefore, 
except at this point, be only two, whilst in it they are four. Such is the obvious 
* By transposing the origin of 9 to the line bisecting the angle TO U, (that is, putting a y/ = s + S, a, = s — S, 
and hence 6 — a. u = Q— e — $ = % — $, and 9 — £ + S = % + $,) we obtain a result in some respects better 
adapted to our final purpose ; but still as the equation so transformed offers insuperable obstacles to a complete 
discussion of the curve, and we have been otherwise able to effect without that aid, it is unnecessary to do 
more than allude to it here. The same may be said of the rectangular equation (76.) itself, when the origin is 
transposed to T, V, M, or U, and referred either to oblique coordinates coincident with the asymptotes, or to 
rectangular coordinates bisecting the angles of the asymptotes, or having one coincident with the magnetic- 
axis itself ; or again, referring the system to rectangular coordinates through O, one of which is parallel to the 
magnetic axis ,- and so on. By one or other of these I have been able to obtain a few properties of the curve, 
but by no one, nor by all of them together, to deduce anything approaching to a complete development of its 
properties, the form of its branches, or its singular points. Could it have been so effected, there is no question 
that it would be the more elegant mode of proceeding, viewed in reference to mathematical symmetry ; and for 
that reason I have spent a good deal of time in attempting it ; but after repeated failures, I am compelled to 
admit that, in the present case, “fallere et fugere est triumphus 
