
_the equations 
‘wherefore the line e=p sin 2V is an asymptote to the curve. 
| asymptote may be found by a very simple construction. 
| Describe a circle round O with the parameter p for its radius, 
| parallel to it in F and J, join EF, E/ and draw parallel to those 
| values of the angle V; those changes may be better shown 
| by diagrams than explained by words; for this the accom- 
panying twelve accurately-drawn varieties have been prepared. 
| The first of these, corresponding to V =50°, shows a symmetric Fig. 2. 
FROM A POLISHED REVOLVING STRAIGHT WIRE. 275 
wherefore the value of OD becomes 
op — (ab+H8) sin 20 + (B—a) 4.00820 

. (a+) sin @ ‘ 
now a+ p= J (a +p"). /(B"+ 4") cos 2VOZ 
(B—a)p= / (a? +7). / (8? +p?) sin 2VOS , 
hei. it. ab sin2VOD 
so that 2 | veer: <n ZOD * 
The fourth proportional to a+ 8,qa and } determines the size of the curve, 
and may be called its parameter, p ; and we may write V for the characterising 
angle VOZ , and so have, more concisely, 
OD=p. sin ae : 
here we may observe that OD. sin@=z,, and thus determine the curve from 
2=p.sin2(V+0); z=a.cotd. 
On eliminating 6 from these equations we get 
2a+a°=p.sin2V (z*—2*)+p. cos 2V. 2a, 
which is of the third order in regard to x, of the second order in regard to z. 
From these equations it is seen that the whole curve is included between 
the two indefinitely extended parallel lines r= +p and «=—p, and that z is 
infinite when §—0, and consequently when a=p. sin 2V; 
The intersections of the curve with a line drawn parallel to the 
cutting the asymptote in E and intersecting the proposed 
OP, Op; P and p are points in the curve. 
The aspect of this curve changes remarkably for different 

curve lying all on one side of its asymptote and without any 
\reflexure. The last of them, corresponding to V =0, shows a circle transfixed 
| by a diameter extended indefinitely both ways. It is interesting to study the 
| transition from the one to the other of these incongruous forms. I have divided 
the half right angle into five equal parts and constructed the curves for 
