Telescopic Mirrors and Object Lenses. 97 
It is possible therefore to form a paraboloid by a regular 
method, by taking first a perfect sphere and grinding it toward 
| 
els 
3 
a= 
- 
S 
SIs 
S 
a 
ae 
Sa 
| 
= 
== 
a 2 2 m* Am 
=aq{it mo yr a ck a oar y— &e.—1} 
peg epee, wai 2 IF ye, 
= iss 3 ys ; y &e.; 
f 3 
similarly, « = r = ar + “2 yo — &e 
Let PP’= 2; .. (w+-2) is the value of 2’ when 7 =y; 
Bees smu ayh th he 2'n 
epene — 7 4 6_ &e 
: yr myt , 2m : 
and by first equation, Br ee 3e = 6 &e. 
~_ 3 — 3 
.. by subtraction, z= ~ m yt— 2 (mi = 4 &e 
m7" 'constant quantity; or zoc y4 ultimately; that is, the thick- 
Therefore the limit of 7 = 
ness, parallel to the common axis, intercepted between the two conic sections varies, for small 
ares, nearly as the fourth power of the distance from the point of contact. Hence, to grind 
a conoid from a sphere, the quantity of friction applied to every point of the surface should 
vary as the fourth power of the distance from the centre of the mirror, and the parts to be 
left on the surface of the pitch tool should be bounded by a curve whose equation is zc< y*. 
The specific conoid will depend on the value of the constant quantity ("*) 2 
EXAMPLETI. (Fig. 7.) Let AV be a parabola; AV’ a circle ‘always. 
In the parabola, yan; «.m=0, 
circle, Ya=axr—a*; ».n=—1; 
m—n 1 y* ; 
pe =p F t= ee the equation to the curve. 
Fig. 5., represents on a reduced scale the whole grinding surface to be left for a mirror 
three inches in aperture, and two feet focus. 
Ex. II. Let AV be an hyperbola; its semiaxes being v and 6; 
. 2 20% b* a. oe — b* 
SE cae ol 
Vol. If. Part I. N 
