CONCERNING TERRESTRIAL MAGNETISM. 
85 
same, and on different sides of the magnetic axis T U, as the slightest consideration 
will render obvious. In order, therefore, to separate these cases and adapt them to 
our immediate subject, we must recur to the magnetic curves themselves, and ex- 
amine their particular characters with more care than has hitherto been done. We 
shall thus be enabled to establish the duality of the vertical points where the centres 
of force are, as assumed in the hypothesis, only two, and of equal intensity. The 
same method, it will readily appear after a little consideration, will establish analo- 
gous conclusions, whatever be the relative nature and intensities of the forces F, 
and F /( resident in the two centres T and U, should, at any future time, such an in- 
vestigation be considered necessary, and prove that in no case can there be more 
than four such points on the earth’s surface. In the present paper it will be shown, 
that could we imagine such an hypothesis to have any foundation in nature, the ex- 
istence of two poles of the same kind and of equal intensities would in certain cases 
produce four points on the earth’s sm-face, at which the needle would be vertical, and 
in others only two. 
XXII. — To trace the Magnetic Curve, and determine the Nature of its Branches and 
Singular Points. 
The equation of the curve* is 
cos 6, + cos = 2 cos (3 . (83.) 
(a.) Let T and U be the poles ; then since the equation is to be fulfilled by the 
cosines of 6, and 9 IP we have the four following systems of equations, each of which ful- 
fils the condition of (83.) for the same numerical value of 0 t and 9 U . (Plate XII. fig. 9.) 
1 . cos 6 t + cos 0 n = 2 cos (3. 
2. cos (— 4) -f cos 9 U = 2 cos (3. 
3. cos ( — 0) 4- cos (— 0 n ) = 2 cos (3. 
4. cos 9 l -j- cos (— d u ) = 2 cos (3. 
(b.) Hence the equation of the cosines determines the four points N 1} N 2 , N 3 , N 4 
corresponding to the four forms of the equation just given respectively; and each of 
these points will trace out a branch of the system of curves as 6 P and consequently 9 U 
is made to vary its actual angular value, cos (3 retaining the same value throughout 
the whole. When 9 t and Q n are both on the same side of the axis, that is both + or 
both — , the branches traced out are the convergent ones; but when on different sides, 
or one + and the other — , the branches are the divergent ones. 
(c.) The four branches thus traced form two pairs of symmetrical portions, viz. 
Nj is symmetrical to N 3 , and N 2 to N 4 . 
(d.) When 6 t and & n have interchanged their values, the four points will have at- 
* See Philosophical Transactions, 1835, p. 238. 
