10 APPLIED MECHANICS 
Substituting these values in the equation (X —a)?+(Y — ap = R?, the 
iaatk Se 
d*y 
da? 
The sign to be taken in the numerator of the right-hand side of this 
last expression should be the same as that of the denominator, so as to 
make the value of R positive. 
If the curvature of a curve is very small, and the inclination 0 of the 
result R= is obtained. 
tangent at any point is small (Fig. 3), then ge tan 0 Y 
dt 
is also small. In this case the expression just found eae ly 
A “bag . * 
for R becomes R=dy nearly or soo and this Fig. 3. 
dx? | 
is sufficiently accurate for the curves into which beams and struts 
deflect. 
10. Construction of Parabola.—A problem of very frequent occur- 
rence is, given the vertex A (Fig. 4), axis 
AB, and double ordinate CBD of a para- C B D 
bola, to construct the curve. Complete the 
rectangle CDEAF. Divide AE into any 
convenient number of equal parts, and 
divide ED into the same number of equal 
parts. Join the points of division on ED | 
with A. Lines through the points of 
division on AE parallel to AB to meet F Att 2.3 € 
the former lines as shown determine points Fra. 4. 
on one half of the curve. Points on 
the other half of the curve are found in a similar manner. 
11. Equations to Parabola.—OK (Fig. 5) is a fixed straight line, 
and F is a fixed point. P is a point which moves in the plane of F and 
OK, so that its distance from F is always equal to its distance from 
OK. The path of P is a parabola, whose axis is the line through F 
perpendicular to OK. The line OK is called the directrix, and the 
point F the focus of the parabola. The curve cuts the axis at A, the 
vertex of the parabola. FA is equal to AO. 
Draw PK perpendicular to OK, and PN perpendicular to the axis. 
Draw the tangent to the parabola at A, and let it meet PK at K’. The 
tangent at A is obviously perpendicular to the axis of the parabola. 
Let FA=a, PN =a, and PK’= 
Then PN? +FN?= adit = ON? 
That is, a? +(y—ayP=(y+a)y. 
Therefore tR=4ay . : ; : i, Os 
which is the equation to the canbe referred to the axis of the 
parabola and the tangent at the vertex, the axis being the axis of y, and 
the tangent the axis of x. 
If the axis of « be moved parallel to itself until it is at a distance 
