80 
MR. DAVIES’S GEOMETRICAL INVESTIGATIONS 
tained symmetrical positions with respect to the line T U, but inverted in respect to 
the extremities T and U, viz. the positions n x , n 2 , n 3 . n A of the figure. 
(e.) The latter system of points is symmetrical with the former, taken with respect 
to Y M Y' at right angles to the magnetic axis, and passing through its centre M, viz. 
Nj with Wj, N 2 with n 2 , &c. 
(f.) The points N 4 and N 3 trace out branches which are continuous from U to T, 
respectively above and below the magnetic axis. 
(g.) The points N 4 and N 2 trace branches which diverge more and more from 
Y M Y 1 as the angles approach to equality, since the angles TN 4 U and TN 2 U 
become more and more acute (being the differences of N 4 T U and N 4 U X, and of 
N 2 T U and N 2 U X, respectively), and when these are equal the points N 4 and N 2 
I)ecome infinitely remote ; that is, these are infinite branches. 
(h.) For the same reason the points w 4 and n 2 trace a pair of infinite branches 
turned from the perpendicular Y M Y' to the middle of the magnet, in the opposite 
direction to the former. 
(i.) The branches are symmetrical to the magnetic axis and to the perpendicular 
through its centre, in the same cases in which the tracing points were severally sym- 
metrical. — The general figure of the several systems of branches is represented in 
Plate XI. fig. 10. 
(k.) No other points than N l3 N 2 , N 3 , N 4 fulfil the equation for the same angular 
values of 0 t and G n ; and hence no other branches than these can exist. 
(l.) Describe the parallelogram T P U P', having T U for its diameter, and the 
angles at T and U above and below the line T U each equal to (3. Through M (the 
centre of T U) draw the lines q 4 M Q 2 and q 2 M Q 4 parallel to the sides of the paral- 
lelogram. Then the infinite branches traced out by N 4 , N 2 , n A and n 2 have these lines 
for rectilinear asymptotes, whilst P and P' are the vertices of the finite branches pass- 
ing from T to U above and below the axis of the magnet. (Figg. 10, 11, 12.) 
(m.) Moreover, if from centres T and U with radii T P, U P circles be described 
cutting the axis in G and II, and from these points lines be drawn parallel to P P' 
uniting the circles in K K' and L L', then the parallelogram formed by drawing the 
radii through these points till they meet in Y and Y', will have its sides tangents to all 
the branches of the curve that pass through T and U respectively in those points. 
(n.) The points T and U are true points of inflexion of the branches, the conver- 
gent ones above being continuous of the divergent ones below the axis, and the con- 
vergent ones below the axis being continuous of the divergent ones above the axis, 
the first series n 2 T n x P Nj T N 2 being marked in full line, and the second in dotted 
line. (Plate XIII. figg. 11, 12.) 
(o.) If we conceive the plane of the curve to be a sphere of infinite radius*, then 
* The idea of considering the plane as an infinite sphere was first, I think, employed by the veteran geo- 
meter Hachkttk, for any real mathematical purpose except that of tracing a mere analogy between plane and 
spherical trigonometry, in his solution of Vieta’s “ Problem of Spherical Tangencies.” This use of it is now 
