50

THE PRINCIPLES OF BRIDGES.

TRACT l.

of ci, this becomes ci =

ar l

(r-jr)

DK X DQ !p« 3

•, which is the

very same expression as the value of ci in the case of the

circle in the former example, and which belongs equally tothe ellipse in both positions, that is, both with the longer axevertical, and with the shorter one vertical, as it is in thefigure to this example.

Hence it appears, that the flat ellipse is more nearly ba-lanced by a straight horizontal back or wall at top, than thecircle is; but the circle more nearly than the sharp ellipse:the want of balance being least in the Hat ellipse, but most inthe sharp one, and in the circle a medium between the two.

example 3.

To determme the Exlrados of a Cycloidal Arch of Equili-bration.

Let dzq be the circlefrom which the cycloidACD is generated ; andthe other lines as before.

Put a = dk, x — dp,and y = cp = ir, asusual; also put r = dqthe diameter of the circle, and z zz the circular arc dz.Then, by the nature of the cycloid, cz is always equal to dz= z ; and, by the nature of the circle, pz is = ^/ rx—xx ;therefore pc or y ( = cz -f pz) is = s + ^frx-xx. Hence y

- k + _z. U..4. z

x x ', but z is =

\/(rx — xx)r—x

-rby the na-

• t *'

f (rx — xx)

ture of the circle; therefore y is = _

V {rx — xx) ''

r — v .. t rx 1

= 2 xf'(rx—xx)’ makm g ic onsfant. Hence

CI IS =

— xy a

~i r ~'

i-ra

'(r-xf

But at the vertex n, x = o, and