THE PRINCIPLES OF BRIDGES.

TRACT 1.

4'i

* 5 q a

yxy^—jr— *y>

where q denotes a certain given or constant quantity, thevalue of which may be determined by making the generalexpression equal to a or dv, the height at the crown of thearch.

Or, the general value of ci is

Corollary 1.—.Because, at the vertex of the curve d, the

CC z

angle of elevation is nothing, or its secant — = j- — 1 the

radius, and the radius of the curvature there being r; there-fore the general expression for the height, becomes there

ci

DV = a = —consequently q a a R, which is the general

value of a for all curves whatever, expressed in terms of theheight a at the crown, and r the radius of curvature at thesame point. Hence then, substituting this value of a in-stead of it, the general expression or value of ci becomes* 3 ok (x* +j l )z an

y r y r

Corol. 2.—Because, in all curves that are referred to anaxis, the general value of the radius of curvature r, is =

z?

j „ ; therefore, by substituting this value for r in the

last expression, the general value of the height ci then be-

yx — xy yx — xy —xy ,

comes —x an = •— r ,— X Q, or = —— x a when x is

y* y> 7 y* «

constant.

For, as either x or y may be supposed to flow uniformly,and when, consequently, either of their second fluxions maybe taken equal to nothing, which will cause one of the termsin the numerator of the above value of ci to vanish; there-fore, by striking out either of those terms, and then extermi-nating either of the unknown quantities by means of theequation to the curve, the particular value of the height Qt