26 MATHEMATICS. 
asymptote. Miller of Groningen has proposed to apply the conchoid to the 
measurement of barrels. 
4. The cardioid ( fig. 125), a curve of the third class, properly an epicy- 
cloid of two equal generating circles. Its equation is y'— (a + 2ax — 22’) 
y— 2ax’ + 2z'= 0. 
5. The lemniscata (fig. 1380), a curve of the third class, discovered by 
Jacob Bernouilli, and investigated by Euler and Fagnano, whose equation is 
(x? + 1) =2@¢ (7—y’). 
6. The ophiuride, discovered by Uhlhorn, for the trisection of angles, 
constructed as follows (jig.131): Construct a right angle, abc, with deter- 
mined sides, ab, bc ; draw from c to any point in the line ab, or its prolonga- 
tion, a straight line, cd; erect at d a perpendicular, dn, to cd, and upon this, 
from the end of the other side of the angle, let fall a second perpendicular, 
am, then m will be a point of the curve. Taking ab =a, be = b, the equa- 
tion of the ophiuride will be z* + (y’— ay) x— by’ = 0. 
7. The scyphoid, according to Uhlhorn, is formed in the following manner 
(fig. 132): If, from any point, 0, out of an unlimited straight line, yy’, a 
perpendicular, 0b, and any oblique line, oc, be drawn to the line, and through 
c a line, nz, perpendicular to oc, and on nz the distances cm = cm'= be, then 
will m and m! be points of the scyphoid. Taking ob = a, then, with o as 
origin of co-ordinates, ob as axis of abscissas, and yy' as ordinates, the equa- 
tion of the scyphoid will be y*— 4a (a—2z) y’— (a —z)'= 0. 
Examples of curves whose equations are most readily expressed by polar 
co-ordinates, are afforded by the spiral lines (pl. 1, fig. 51), which wind 
continually around a fixed point, either continually approaching to, or 
receding from it, according to a given law. The simplest of these is the 
Archimedian or equable spiral (pl. 3, fig. 183), which is generated when a 
point moves uniformly along the radius of a circle, this radius describing an 
uniform rotation around its extremity, so that the distance of the moving 
point from the centre is always proportional to the angles described by the 
radius. It is generally provided, in addition, that the moving point shall 
meet the circumference of the circle by the time that the radius has 
described its first entire revolution. 
Spirals may also be described on the surface of a cylinder, a sphere, or a 
cone: the well known screw line (pl. 1, fig. 52) belongs to the cylindrical 
spirals. 
The cycloid or trochoid belongs to the transcendental curves. This is 
described by a point in the circumference of a circle which rolls along 
a straight line until it has completed a revolution; the circle, curve, and 
line, being supposed to continue in the same plane (pi. 3, fig. 1385). If the 
revolution be started when the point lies in the straight line (at a), and is 
consequently the point of tangency between the circle and line, and 
continues until it again meets the straight line (at A), then the line Aa, 
called the base of the cycloid, will be equal to the circumference of the 
generating circle. The cycloid cuts the base at A and a’, therefore the 
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