98 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
curve becomes infinite, it coincides with the axis Z' Z (XXVI.), and hence the final 
tangent must also have a point of coincidence at an infinite distance from M with the 
line Z' Z. Nor can the curve meet the line Z' Z in any point to the right of U, and 
at a finite distance from it. Since, if it can, the convergent branch of the magnetic 
curve also meets the axis at that point. But the tangent to every finite branch of a 
convergent magnetic curve makes a finite angle with Z' Z at U, and having no points 
of inflexion, it cannot meet the tangent again. But if it meet Z'Z, it must have pre- 
viously crossed its tangent at the point U. Hence the hypothesis of the curve of 
vertieity meeting the axis Z' Z at a finite distance involves a contradiction. The 
magnetic axis Z' Z is therefore an asymptote to this branch of the curve of vertieity. 
Precisely the same circumstances take place in the branch lying above the axis and 
to the left of T. The convergent curve has therefore two asymptotic branches, the 
line of the magnetic axis being the rectilinear asymptote to them both ; and no other 
points of the curve lie on the opposite side of that line from O. 
In the next place, for the determination of the branch or branches of the curve 
lying on the same side of the magnetic axis with O : — 
For the same reason as before, the line O Q' cannot be a tangent to any one of the 
curves lying below the axis ; and the first curve that can have a tangent is that whose 
tangent at U coincides with T O ; and as the same line is a tangent to the curve above 
and to the curve below the magnetic axes at their common point, the branches above 
and below arc continuous ones. 
For any other position, as O N, there is always one convergent magnetic curve 
which can touch it, as at N' ; and the two distances O N and O N' are the two values 
of r, which correspond to any specified value of 0, as V O N, in the general polar 
equation of the curve (82.). The point N' will hence trace out another branch of 
the convergent curve UN' II (H being determined as already specified) corresponding 
to the asymptotic branch to the right of U, which branch will be finite, and comprised 
within the rectangle O V U L. Also, since by (82.) there are but two values of r cor- 
responding to each value of 0, there are no other branches to the right of M H besides 
these two. 
Divide TU produced in S', in the triplicate ratio of T O : U O, (or S'U^S'T 3 :: 
() U : T U), then O S' is a tangent to the curve which passes through O. No curve 
which passes more remotely from the axis than O can have a tangent drawn to it 
from O, since O is on the concave side of all such curves, and they have no points of 
inflexion. Nor can curves passing through T have tangents drawn to them, for 
reasons before given, till /3 has become such as to render OT a tangent at T. From 
that state till the curve passes through (), the point Olies on the convex side of them, 
and hence from O two tangents can be drawn to each individual curve, one on each 
side of O, but which coalesce in the single tangent at that point. The curve is hence 
continuous from U to (), and from T to O ; and since they have at that point a coa- 
lescent tangent, they form continuous branches not interrupted at O, their point of 
