THE CUBIC PARABOLA. 57 
tan ¢ = 3 tan #6. 
| tan 6 tan d 
4 a = ye SATE Se 
Jom peers 1 
Length of Parabola 
y= moi. m = —, 
dy ae ae 
is = 3m 2x 
ds / dy \2 
tat Sg tte) 
Therefore $s age & af 1 +9m2a2*t de 
Expanding, and integrating the individual terms we find 
8=2+ +5; m’ «* — $m* x” ete Qa 
Remembering that m = 
we find 
2 4 
reer E 2% 
Dividing both sides of equation by x we obtain 
© ih cided te 
Remembering that tan 6 = 4, we have 
s=a (1+ 5% tan?0— 2 tan*d...) 10 
The first two terms of any of the above expressions will give 
the length of the curve approximately. In the computation of 
the tables, the third term has been taken into account. 
The area A contained between the co-ordinates, «, y and the 
curve is easily obtained. 
Area A = f/ydx = /m2° dx 
, integrating, and remembering that m = —";, we find that 
9 A=2¥ it 
oy 4 
‘The area therefore equals, one fourth « multiplied by y 
Length of Circular Are. 
The length of circular arc CC’ is obtained in the following 
manner, (See fig 2 
