CONCERNING TERRESTRIAL MAGNETISM. 
89 
We may now establish the truth of the construction given at (p.) of art. xxn. for 
finding these points. 
Let P T = radius = 1 ; then P M = sin (3, and LN = ^=. Hence T L = cos (3 
— and L U = cos (3 + are the corresponding values of cos 6 t and cos Q lt in 
equations (89.) : and the intersection, therefore, of the lines B U and D T give one of 
the points in question. 
In the same manner, by taking the other combinations of + and — as signs of 
and 6 U in these equations, as signified in the opening of the last section (xxii. a.), 
the other points will be shown to be those given by the construction enunciated in 
(xxn. p.). 
It is also clear that the limitation to which the construction is subjected is that 
expressed by the limitation of the equations themselves. 
Moreover, the distribution of the points as to the particular branches of the curves 
to which they belong is properly made : for T n x , U n x are the radiants belonging to 
values of and 6 n on the same side of the axis. Hence n x is in the convergent curve. 
So, for the same reason, are n 3 , N 4 and N 3 . Again, since n 4 is found by the intersec- 
tion of radiants n 2 U and n 3 T, whose corresponding values of cos and cos 6 tl are 
estimated for and on different sides of the axis, n 4 is in the divergent curve. So 
also, for the same reason, are n x , N 4 and N 4 in the divergent branches. 
XXIV. — The Points of Intersection of the Finite with the Infinite Branches: and on 
the Asymptotes. 
1. The construction of the points of intersection has been given in (xxn. q.), and 
it is thus proved. (Plate XIV. fig. 13.) 
When in the construction of the points, the two points N 4 and N 4 coincide, the 
radiants N 4 U and N 4 U coincide also. But N 4 UX = TUN 3 — TUN 4 . Hence 
when the radiants coalesce they form a line at right angles to T U. 
Also in this case cos 6 n — 0, and the equation of the curve becomes at this point 
cos 6, = 2 cos (3. 
But taking r l = 1 , T M = cos (3 and T U = 2 cos (3 ; and if G T be drawn it is = 1 . 
Hence the construction is true. 
2. The construction of the asymptotes has been given in (xxii. l.) ; and its truth 
may be thus established. 
To prove that q 2 M Q 4 and q 4 M Q 2 are Asymptotes to the Infinite Branches. 
(Plate XIV. figg. 11, 12.) 
Since the point of the curve is found by the intersection of two parallel lines PT 
and P' U, it is infinitely distant ; and the two radii r t and r u themselves being infinite, 
are equal to one another. But the tangent to any point of the magnetic curve divides 
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