CONCERNING TERRESTRIAL MAGNETISM. 
101 
drawn from O to each curve, and trace out segments coalescing at O, where the two 
tangents coalesce in one, and the segments therefore also being continuous. Portions, 
then, of the curve of verticity will lie on both sides of M O in this region : one por- 
tion in the angle TMO between R and O, and the other in the angle O M Y\ 
Now when the asymptote to the magnetic curve passes through O, we have seen 
that there is a tangent to it at R. But in the same case, M O P" is itself a tangent to 
the same branch at a point infinitely distant. As the magnetic curves approach to- 
wards O, there will still be two tangents possible, the point of contact of one becoming 
continually nearer to O in the angle P"M Y, and the other in the angle R M P", till, 
as before shown, they coalesce continuously in O. The branch lying in the angle 
P" M Y', therefore, has M O P" for a rectilinear asymptote, as the branch in the oppo- 
site region had M P. 
It would probably be difficult to establish, from any considerations furnished by 
the properties of the magnetic curves, the utmost angular extent to which the infinite 
branch lying in the angle O M Q extended from the asymptote M O P". But recur- 
ring to the fact furnished by equation (82.), that for every value of d which gives one 
real value of r there is also a second real value ; and as for all values of 6 comprised 
between and a lp or within the angular region T O U, we have shown that there is 
one real value of r ; and, with the exception of the angular region M O W, (W O W' 
being the tangent to the magnetic curve at O,) we have established two values ; it 
follows, that for completing the whole series, and fulfilling the conditions of (82.), 
there must be a second value of r for every direction which a line can take in the 
angle P" O W' ; or, which is the same thing, the infinite branch will touch the line 
O W f , but can never pass to the right of it ; that is, it lies wholly in the angle P"OW', 
and never meets the line O W' again after it passes through O. 
The course of each curve of verticity is thus fully made out : and it now appears 
that each is confined within specific and peculiar angular regions referred to lines 
drawn from O through the magnetic centres of force. Though in itself the divergent 
system is not required in the present physical problem, yet the separation of them as 
constituents of the equation (76.) was essential to enable us to ascertain the separate 
branches of the convergent one, which we have yet to discuss in its application to 
that problem. For the more complete and ready understanding of the whole system 
of branches, I refer to the figure of them as the representation of the complete equa- 
tion (76.), the dotted branches representing the curve of verticity for the divergent 
branches of the magnetic curve, and the full-lined ones that for the convergent 
branches, to the consideration of which we shall hereafter return*. (Plate XV. 
fig- 17.) 
* the cases examined, the point O was without M H : but when it is in that line the divergent curve has 
its asymptotic branches both approaching the asymptotic on the same side of O, viz. on that on the opposite 
side of f U from O. The figure (Plate XVI. fig. 18.) will render further verbal detail unnecessary here. 
