An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE CATENARY.

229

from which we obtain
d- tan r = sinh x/c,
and therefore y =c cosh /c.............................. (2),
which is the Cartesian equation.
In the figure let OG be a horizontal line such that OC= c;
then OG is called the directrix of the catenary. Let the tangent,
normal and ordinate at P meet OG in T, G and M. Draw MS
perpendicular to PT. Then
dy 1 dy
tan. = dx c dk cos f,
whence y = c sec *.
But y=PM=MSsec,
therefore 1MS = c.
Also s = c tan r = MS tan r = PS,
which shows that the locus of S is the involute of the catenary.
'Again if p be the radius of curvature at P,
ds
P = sec2 r
= PM'sec = PG,
whence the centre of curvature of P is at a point O' on GP
produced such that PO' = PG.
If a catenary revolve about its directrix, the line PG is the
radius of curvature of the circular section of the surface of revolution thereby generated, whilst PO' is the radius of curvature of
the meridian section; and it is known from solid geometry that
these are the two principal sections of the surface. Hence the
surface generated by the revolution of a catenary about its
directrix belongs to the class of surfaces which have their two
principal radii of curvature equal and opposite.
346. It can also be shown by the method of ~ 314 that if a
parabola roll on a straight line, the locecs of its focus is a catenary.
If however the conic is a central one, the locus of the focus is
a more complicated curve*.
* Besant, Notes on Roulettes, p. 47.