56 Cc. J. MERFIELD. 
The co-ordinate y varies as the cube of the distance A: tho 
axis of X. 
Angled =CDB=CFG 
tan d = 3m 2,? 
= EB = R Sin ¢ 
y =GT=R- RCos¢ 
H=TH=y,-GT 
EO=2, — 2 : 
The preceding investigation is an exact method of locating the 
cubic parabola, as applied to the easing of circular ares. . 
Before explaining an approximate method, we will deduce some — 
useful formule from the equation to the curve. 
y=m saa 2' 
tan ¢ = =3mz2,? 3 
From equation 2’, m = = ; and by substituting this value of m 
in equation 3 we have i 
3 
tan ¢ = — 6 
Therefore the tangent at C, if produced to D, makes an angle 
with the axis of X, the tangent of which equals ove Further — 
than this we see from equation 6, that D B= 4 xc, and O D=3te 
Again DC = y, Cosec 
Tec 
Cot 4 ~s 
Therefore DC = Ye/ Lee, os v 208 ed Se § a 
Let the angle contained nitheae reg radius vector, and the 4 
axis of Y be denoted by @. 
tan 9 = 
¥=me* 
Therefore tan 6 = m x?.. 8 
The tangent of the sates varies as the square of the distanc® — 
along the axis of _Y. 
tan ¢ = 3 m 2? 3 
eee eee 
Comparing this last equation with number 8 we have 
