CONCERNING TERRESTRIAL MAGNETISM. 
99 
union. The branch T O H is therefore also finite, and corresponds to the asymptotic 
branch to the left of T, as U N' IT did to the asymptotic branch to the right of U. 
7 r 7 r 
In all these cases, and for all values of 6 from a, to — and from a to + 
we have therefore found two values of r, and by equation (82.) these are all that can 
exist. Nor can any other tangents than those we have described be drawn to the 
convergent system of magnetic curves ; and hence it appears that the general equation 
(82.) applies to the convergent curves of verticity for no other than the values above as- 
signed. 
XXIX . — On the Branches of the Curve of Verticity corresponding to the divergent 
System of Magnetic Branches. 
Let O, T, M, U, and Z' Z denote the same things as in XXVIII. ; and draw the line 
O T, O U, and O M, which continue indefinitely. (Plate XIV. fig. 15.) 
Then since each individual divergent curve has an asymptote passing through M 
(XXIV.), the line O M is an asymptote to some one curve ; and as the asymptote is 
itself a tangent from O to that curve at a point infinitely distant, the corresponding 
point in the divergent branch of the curve verticity is itself that same point. Whence 
the line O M is an asymptote to the branch of the curve of verticity ; and since the 
asymptote itself lies in the angle Y M U, the branch to which that asymptote belongs 
emanates from U, or otherwise the magnetic curve must have crossed the vertical M Y. 
There is also one curve which can touch O U at U, determined, as before explained, 
by the value of (3 ; and there can be no one drawn between this and the produced 
axis U Z, which admits of a tangent from O ; for in that case the curve is concave 
to O, and has no points of inflexion in that branch. 
Take some intermediate position, as ON; then since the point O is on the convex 
side of the curve U N, a tangent can be drawn ; and only one for the curve is convex 
to MY, and has no points of inflexion in its finite branch. Nor again, can it be 
touched in the other part of the branch by a tangent from O, since O is on the op- 
posite side of the tangent U T at the point of inflexion U. Also, as this is the case 
for all positions of the asymptote within the angle P M U, the series of contacts will 
trace out a curve, commencing at O, and having O M P for an asymptote ; and there 
is only one branch situated within the angle U M P. 
Again, let the curves be continued, whose asymptotes lie within the angle P M Y, 
as M P. Then any line from O to the left of M will cut the asymptote before it 
meets the curve, and hence cannot be a tangent to that curve. None of the points 
of the curve of verticity corresponding to the divergent branches of the magnetic 
curve are therefore situated within the angle P' M Y. 
Attending next to what takes place in the angle U M Y', we see that the first curve 
that can have a tangent drawn from O is that which has O U for its tangent : for any 
other would lie on the opposite side of its asymptote from O. 
o 2 
