86 THE PRINCIPLES OF BRIDGES. TRACT I.

aa—yy „ n aa — 4?/?/

-^ X j, the fluent of which is-x y : and there-of# aa J

fore the force on the base is to the force on the circular end,

aa — iyy

as y is to-- x y, or as aa to aa — \yy, or as 3 aa to

3 aa — yy. And when y = a = Ac, the proportion becomesthat of 3 to 2. So that, only one-third of the absolute forceis taken off by making the end a semicircle.

Corollary 3.— When the face adb is a parabola.

Then, the notation being as before, viz, nc = a, and ac= b, it is a : x : : bb '.yy ; hence x = and * = ~~bb~>which being written in the general expression, the fluent ofit becomes the circular arc whose radius is — and tangent y ,

or = — x arc whose radius is 1 and tangent -5-; so that2a ° bb

the absolute force is to the. force on the parabolic end, asy

, ,• M , .

is to the arc whose tangent is y and radius — ; that is, as thetangent of an arc is to the arc itself, the radius being to thetangent, as 1 to or as 2 to -j^. And when y — b, the ra-

tio of the tangent to radius, is that of 2 to — : or that of 2

to 1 when dc = ca. In which case, the whole force is tothe force on the parabolic end, as the tangent, which isdouble the radius, is to the corresponding arc ; that is, as thetangent of 63° 26' 4" to the arc of the same, or as 2 to l - 10714;■which is a less force than on the circle, but greater than onthe triangle. And so on for other curves; in which it willbe found, that the nearer they approach to right lines, theless the. lorce will be, and that it is least of all in the triangle,in whicu it is onc-hulf of the whole absolute force when right-angled.