92 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
cates the opposite branches to those just described: and hence the same remark may 
be made respecting them. 
The fourth and fifth equations are generally inconsistent with the equation of the 
curve, and hence in all those cases imaginary. When, however, cos (3 = 0, or (3 = 
the equation of the curve is consistent with this equation. The divergent branches 
in this case respectively coalesce with the magnetic axis itself, and the convergent ones 
by their continual expansion outwards have then come to coalescence with the axis 
produced ; each through its whole length. In these cases, appoint in the magnetic 
axis may be considered as a point of inflexion, and the tangent in all cases so taken 
makes an angle 0 or n with the axis. 
When the middle M, however, of the axis, and its opposite point on the infinite 
sphere are taken, any line through them may be considered a tangent, as the 
asymptote, properly speaking, has then ceased to exist, or to be expressed by the 
equation. In other words, the direction of the curve at these points is become pro- 
perly indeterminate. That the expressions themselves indicate this, will be made to 
appear in the next section. 
XXVI. — To fnd the Multiple Points and the Directions of their Tangents of the Mag- 
netic Curve. 
At a multiple point we shall have, in consequence of (83.) and (85.), the three 
equations 
1 . (sin 3 6 t — sin 3 0„) sin sin 6 U sin 2 6 : -J- 0 n = 0, 1 
2. (sin 2 0, cos, + sin' 2 0 tl cos 0 U ) sin 0 I sin 0 n sin 2 d t + = 0, j* • • • (97-)* 
and 3. cos 0 / + cos = 2 cos j3. J 
If these three equations be simultaneously fulfilled by the same values of and 6 n , 
the points indicated by those values will be multiple points. But this is the case with 
either of the values 
1 . sin 0, = 0, “\ 
2. sin d u — 0, >- 
3. sin 2 d + 0 U = 0, - 
(98.) 
that is, at the poles and at the point of symptotism ; that is, at the point on the infi- 
Sincc 
