a 
PRELIMINARY ll 
b from the vertex (Fig. 6), then y in (1) will become y+, and the 
es — equation will be 
BG Basay+0)=sayttab 2... (8) 
ae 
It the axis of y be moved parallel to itself until it is at a distance ¢ 
v y i ec 
‘ b x Pp b 
Fo ; Fe 
¥ t <_< 
A 
Fra. 5. Fig. 6. Fa. 7. 
from the axis of the parabola (Fig. 7), then « in (2) will become z+¢, 
and the new equation will be 
(x +c)? = 4a(y +5) . é ; ‘ #3) 
“In the foregoing equations y is positive or negative according as it is 
measured above or below the axis of x, and x is positive or negative 
according as it is measured to the right or left of the axis of y. 
_ 12. Cycloidal Curves.—If a circle be made to roll along a line, and 
remain in the same plane with the line, a point on the circumference of 
the rolling circle will describe a cycloidal curve. The line upon which 
- the circle rolls is called a base line, a directing line, or a director, If the 
base line is a straight line, the curve described is called a cycloid. If 
the base line is a circle, the curve described is called an epicycloid or a 
id, according as the generating circle rolls on the outside or 
inside of the directing circle. 
_. The hypocycloid becomes a straight line passing through the centre 
Fia. 8. Fig. 9. 
of the directing circle (Fig. 8) when the diameter of the rolling circle is 
equal to the radius of the directing circle. 
The same hypocycloid may be described by either of two rolling 
circles whose diameters are together equal to the diameter of the direet- 
ing circle (Fig. 9). 
The same epicycloid may be described by either of two rolling circles 
whose diameters differ by an amount equal to the diameter of the 
directing circle (Fig. 10) 
BLEASE RETURN TO 
WEpT. ‘oF RPPETED (MECHANION, 
