CONCERNING TERRESTRIAL MAGNETISM. 
97 
which is fulfilled either by 6 = a / or 6 = At the points T and U, then, the two 
values of r for such values of 6 as lie to the right and left of these points coalesce, 
and the radius becomes a tangent to the branches of the curve that meet there. In 
the same way for values of 6 between these points the branches of the curve coalesce, 
and the radii- vectores form at those points tangents also to these branches of the system. 
It also immediately appears that there are no other real values of 0 which fulfil the 
condition, and hence there are no other such points besides those now determined. 
By inserting these values of 6 in (91.) we have in the two cases respectively, as we 
should anticipate, r — b sec a p and r — b sec a n for the values of r. 
XXVIII. — The Branches of the Curve of Verticity corresponding to the Convergent 
System of Magnetic Branches. (Plate XIV. fig. 14.) 
LetT, M, U be, as before, the poles and the middle of the axis, and O the centre of 
the earth. Draw X' X through O parallel to T U and O V, or Y' Y perpendicular to 
it. Also draw M H parallel to O V, and the lines O T, O V, which produce to Q' 
and Q respectively ; and confining our attention to one side of the middle of the 
axis M, the corresponding branches may be thus investigated. 
There is always one particular convergent curve to some parameter (3 which will 
touch the line O U at the point U, and which, since O and U are given points, is de- 
terminate and single (XXII. XXIII.). The same is true of OT; and the corre- 
sponding values are cos (3 = sin 2 \ a l and cos (3 = sin 2 \ a ir 
Those curves only in which (3 enters as a larger angle, or in which cos (3 diminishes, 
can have tangents drawn to them from O. For if any line O S' Q' be drawn from O 
between O T and O U, or to cut the magnetic axis itself between the poles, the curve 
being wholly concave to O Y, it will intersect the curve at Q' ; and since there are no 
points of inflexion (XXV.) between T and U in the branch U Q', it cannot again 
meet the curve, and consequently cannot touch it. Hence only the branches of those 
convergent curves which depend on a parameter (3 greater than that already specified 
can be touched by lines from O. 
No point of the convergent system of curves of verticity can therefore lie in the 
region Q' T U Q. 
Let any curve U N be taken to the right of the line O U and above the axis. Then 
since the curve is convex (its tangent U R at U making a less angle R U Z with the 
axis than O U Q) to the point O, a tangent can be drawn from O to a point N in it. 
And since there are no points of inflexion in the magnetic branch, there can be only 
one tangent so drawn to that branch. 
The points of contact, or the points of this branch of the curve of verticity, are 
always at a finite distance from the line of the magnetic axis. For whilst the mag- 
netic curve itself is finite, all its points are at a finite distance from the axis Z' Z ; and 
hence all the points of contact of lines from O to it, which constitute the curves of 
verticity, are also at finite distances from that line. Moreover, when the magnetic 
MDCCC XXXVI. O 
