98 Mr. Cecit on a Machine for Grinding 
the circumference, according to the law above mentioned: 1. e. 
as the fourth power of the distance from the centre of the mirror. 
The quantity of friction applied to the surface of the mirror may 
be made to vary according to any direct law of the distance 
from the centre of the mirror by cutting away certain parts from 
the face of the polishing tool, which is formed of pitch, covered 
with an oxide of iron or tin. The surface of the pitch tool 
after removing the superfluous parts, is represented on a scale 
half the real size, in Fig. 6; which is adapted to a three-inch 
aperture with a two-feet focus. This figure, cut out in a piece 
of pasteboard or tin, may be applied to the pitch surface and 
traced upon it. To prevent the pitch from extending by pressure 
into the void spaces, those parts may be cut away or depressed 
in the tool itself, as well as in-the pitch surface. 
In the circle n=>—1; 
bt 
al — 
_ mon aie v7 +67 | 
SS eae SS ae aS ee 
v? +b? 
ee FS 
the equation to the curve to be traced on the pitch tool. 
Ex. III. Let AV be an ellipse; 
ne, 2b" Pas 5? 
Saree maa cre we n=—s4- 
In circle nm =— 1; 
b? 
les iA 2 2 
m—n v v— Bb 
‘orp ae are 
v—b 
z= a jah 
Therefore, if the point under consideration is the extremity of the major axis, v is greater 
than 6, and the construction is possible; but if the point be at the extremity of the minor 
axis, the construction is impossible. 
If the original figure be a paraboloid with latus rectum (A), and it be required to form it 
into a paraboloid with latus rectum (a), it may be shewn nearly in the same manner that the curve 
< . Yl 
traced upon the pitch tool must be a common parabola with latus rectum ( a —*) . 
