CONCERNING TERRESTRIAL MAGNETISM. 
87 
the dotted and full-line branches meet at the opposite side of the sphere from M ; and 
if we call this point M', the point M' is also a double point of inflexion, and the whole 
system of branches may in this sense be said to be continuous. 
(p.) The following construction gives the vertices of the curve with respect to the 
line P M P'. (Plate XII. fig. 10.) 
Describe an equilateral triangle L P L', whose perpendicular is P M, and from 
centres T and U two circles PKP' and P K' P'. Through L and L' draw lines parallel 
to P P', meeting the circles in A, B, B', A', and C, D, D', C\ Draw radii to the several 
points of intersection of the lines A A' and C C' with the circles ; these will intersect in 
the points n x , n 2 , n 3 , rc 4 and N 1? N 2 , N 3 , N 4 , which are the several vertices sought. The 
outer points n 1} n 3 , N 4 and N 3 are those at which the vertices of the convergent branches 
are situated, and the inner ones n 2 , w 4 , N 2 , N 4 are those of the divergent branches. 
When the points L and L' fall without the double segment P K P' K, the construc- 
tion fails, and there are no such points in the curve when this takes place. This ob- 
viously will be the case when L and L' fall between the poles T and U* *. 
(q.) The points in which the convergent and divergent branches intersect are found 
as follows: — (Figg. 10, 13.) 
Draw through T and U the indefinite perpendiculars to the axis, and with centres 
T and U describe the circles P G H P', PEFP' cutting the perpendiculars in G, H, 
and E, F, respectively ; then E, F, G, H are the points sought. 
When T P is less than T U the construction fails, and the existence of such points 
becomes impossible. When T P = T U, the points coalesce in the poles, and the 
branches all touch there. 
We shall now proceed to establish the truth of such of the preceding properties of 
the curve as are not immediately evident. 
XXIII. — The Vertices ; or the Points at which the Tangent is perpendicular or parallel 
to the Axis. 
That it may be more easily effected, resume the general equation (38.), the poles 
being endowed with equal absolute intensities of force. 
dy __ 
y y 
sin sin 
rf 
r f 
. x 
x H- a 
x — a 
cos 6, cos 6 tl 
o -f- o 
rf r,r 
(84.) 
'ufficiently familiar to the Continental geometers ; but I am not aware that any further use of the principle has 
been made even by them. The application here made of the idea was suggested to my mind several years 
ago in considering the nature of infinite branches in a case analogous to the present one ; and it seems to supply 
a desideratum, the want of which all writers on the higher geometry have felt when discussing the characters of 
certain particulars respecting curve lines. 
* These considerations will be rendered subservient to an investigation of the state of the forces in what is 
commonly, though very improperly, called a “ suturated” bar-magnet, which will hereafter be laid before the 
Royal Society. 
