82 PROCESS OF POLISHING AND FIGURING 18-IN. GLASS 
3. By raising the temperature of both glass and polisher, and 
before the pitch becomes of its usual hardness to use a 
few long strokes (half strokes), afterwards gradually 
decreasing them to nothing. This I have tried with 
partial success. 
4. By local polishing, as adopted by Lassell ; perhaps the one 
now mostly used, and the process by which the greatest 
success has been obtained. Its defect is that smal 
irregularities are almost impossible to avoid. 
5. By graduating the pitch polisher, which in my experiments 
seems to be the process most certain of success ; yet in 
large surfaces, where a considerable amount of correction 
is to be performed, great care is necessary to avoid it 
running into an irregular curve. 
As in this process the main point to be considered is the correct 
system of graduations to be used, I began by inquiring into the 
form of the solid interposed between the sphere and paraboloid of 
the same curvature at the point of contact, seeking thus to com- 
bine theory and practice. 
_ The general equation to this solid becomes complicated, but as 
it was only required to know the variation in the thickness of a 
section from centre to edge by combining the equations of the 
circle and parabola, I deducted an approximate expression (correct 
for the usual shallow curves to eight places of decimals) thus :— 
WANE 2 fake 
Equation to parabola origin A is «' = y y a yy 
ar. 
” circle r= TF — Jr sine Ye 
2 4 6 
« . . oe y y oF gee 
which, being expressed in series, is « =~) + aT T6P 
Let now y' = y, and neglecting higher powers of y than the 
4th, we have by subtraction (a — x') = 2 equal to thickness for 
any value of y = 5 and supposing 7 constant, it is seen that such 
varies as 4th power of semi-diameter. 
Let now z be calculated for intervals of 1 inch in the length 
of y with radius (r)=320 inches as in the speculum under con 
“ema 
