96 Mr. Cecit on a Machine for Grinding 
emery; and lastly, the surface is polished upon woollen cloth filled 
with putty. When the lens is finished on one side, it is separated 
by striking the block edgeways with a hammer: It is again ce- 
mented upon the block, and the other side finished as before. The 
detail of this operation is omitted for the sake of brevity. 
As the parabolic curve is generally considered necessary for 
the object mirror of a reflecting telescope, we will now shew in 
what manner it may be attempted. The common method of grind- 
ing paraboloids is altogether tentative, and depends en the skill 
and experience of the workman. A _ perfect sphere being first 
obtained, it is ground away toward the centre, by a few circular 
strokes upen the polishing tool; till the centre and circumference 
are found to reflect parallel rays to the same point. It may then 
be supposed that the whole figure more nearly resembles a para- 
boloid than a sphere. But a method founded upon calculation, 
especially when improved by practice, may produce a more perfect 
figure without requiring any manual skill; and such a method 
may be adopted on the machine already described. Ifa paraboloid 
and a sphere be made to touch internally at the vertex of the 
paraboloid, where they have the same curvature, the thickness 
intercepted between them, measured parallel to the axis, varies, 
for small ares, nearly as the fourth power of the distance from 
the vertex of the paraboloid. The same law holds also for the 
other conoids, affected only with different constant quantities*. 
* Let AV, AV’ (Pig. 7,) be two conic sections, having the same axis 4Z, and the same 
curvature at their common vertix (4); and let their equations be 
y=ar+ma’; y?=ar'+ma?; 
in which (@), the latus rectum, is common to both, their curvatures at the vertex having been 
assumed equal. 
From the first equation, 
fa +4amy*—a 
oma 2m 
