ica ag Mt ey 
Tog cae 
THE CUBIC PARABOLA. 55 
in which m = . It is in this form that we will use the equation, 
to deduce the formule necessary for the location of the curve. 
In the equation to the curve, the constant “m” may be fixed 
at pleasure, according to the requirements of the case. Now it 
is necessary that the radius of curvature at some point, shall be a 
particular value, so that we must give to the constant a numerical 
value, so that when placed in a particular equation, the desired 
result will be obtained. 
The radius of curvature in any curve equals 
d 
f {1 + sty" (3) 38 
i da? 
Cubic Paiabola y= mx 
oy = om ee 3 
d= 4 
2 Behar 6 Mm 2 
da : 
Therefore p = (149m? 24) : 
6me 
From this equation we may find m, if we know p and a, by a 
method of approximations, until the correct value is obtained. 
The method adopted is as follows :— 
tan ¢ = £4 3 m ae? 
eee. CNet ater ealgy 
6m tan 
Reducing we ep 
5% = Sin $ Cos? ¢ cieseeeenes S 
_ A value of the acer ¢, such that its sine multiplied by its cosine 
_ “Mared, will equal 5% can soon be found. If the tangent of this 
angle be placed in the equation 
tang = 3mz 
We obtain the value of m required. (See table 5.) 
Produce the arc CC’, which has the same radius as the curve 
*t C, to the point 7, until its tangent becomes parallel to the axis 
of Zz, (See fig VY 
CB=y, =m ae? 
