HIGHER GEOMETRY. 20 
foci; the shortest (called minor axis) is perpendicular to the former and 
bisects it. 
The distance from a focus to the centre is called the eccentricity (in the 
circle = 0); the equation of the ellipse is y= ze (a’— x"), where a and b 
a 
are the semi-major and minor axes. In the circle a = J, therefore, y? = a’?—2’ 
is the equation of the circle of radius, a. 
A hyperbola is produced when the intersecting plane is parallel to 
the axis of the cone. As this intersection always meets the base of 
the cone, the hyperbola is an open curve. It also has two foci, the 
difference of whose distance to any point in the circumference will always 
be the same. It is composed of two equal parts, each of two branches, 
which, stretching into infinity, approach continually without ever meeting 
two straight lines (the asymptotes) which intersect each other in the centre 
of the major axis. The equation of the hyperbola is y’? = (ee ay: 
Gi 
When a=), it becomes y? = x*— a’; such a hyperbola is called equiva- 
lent. The asymptotes of this form a right angle with each other. 
The parabola is produced when the plane of intersection is parallel to the 
side of the cone; it also is an open curved line, but has only one focus. 
Every point of the curve is equally distant from the focus and a fixed 
straight line called the directriz. It also consists of two symmetrical, 
infinitely extending branches, which unite in a point half way between the 
focus and directrix, called the vertex. A straight line drawn through the 
vertex and the focus is called the axis. The equation of the parabola is 
y’ = pr. 
The following algebraic curves may be mentioned in addition : 
1. Parabolas of higher orders. These are curves in which a power of 
the ordinate is proportional to some other power of the abscissa: their 
general equation is y*— az". Ifn—1 and m= 2, the equation becomes a 
quadratic (thus, y’—az is the same with the common or Apollontan 
parabola) ; a cubic when m=3,&c. The parabola of Neil (pl. 3, fig. 124), 
whose equation is y? = az?, is particularly remarkable. It is that curve 
in which a heavy moving body falls equally in equal time. 
2. The cissoid (fig. 122), a curve of the second class, discovered by the 
Greek geometrician, Diocles. It consists of two infinite branches, uniting 
in a point, a, and continually approaching a tangent of the circle (the 
asymptote) without ever meeting it. Its equation is z°= (a—z2) 7’. 
3. The conchoid (pl. 8, fig. 123), a curve of the third class, discovered by 
Nicomedes, whose equation is oF + y’=a’. Its construction Is very 
simple: draw a straight line, and out of this line take any point, a; from 
this point draw a straight line cutting the firstin g; from q take off gm = qn 
in this second line equal to a given or fixed length: m and m will be 
points of the two infinite branches of the conchoid, which also has qq for its 
Qe 25 
