for Optical Purposes. 201 
and the differential of this with regard to an arc of the circle 
must be zero. Differentiating and reducing by the equations 
dx _ y—y' db _ p 
dy x—x / dy G(x—a/) 
we have 
p{2^(y-/)-2^^-^)-g[66 2 -(^+/ + a 2 )]} 
+ ^My-/)-y(*-*0]}=o. 
It is more simple to express this result in terms of E, r, p 
and the angles between them. 
Let yti be the angle between p and r, and v that between p 
and E. Let us also put 
Let /3, y, 8 also represent the angles made by r, E, and p 
respectively with the line joining the source of light and focus, 
and let 
Then we have 
_ Ecos7 + pcos/3 _ Esiny + rsin/3 _rcos/3— EC0S7 
*- - — g » 3/- 2 ' a 2 ' 
{b 2 -a 2 ){y-yj + b\x-a/f=p\b 2 -a 2 sin 2 8), 
6 2 — a 2 = Ercos 2 a, 
E + r . r-E 
smn= — = — sin*. cosi7 = — r — cosa, 
2a 2a ' 
E=& — j-«, p = b+ -rX, 
, cos v sin 7 sin /3 Er . 
#=6 — , w = a ? = -p- sm 77 cos «, 
cos a sm a cos « 6 
b 2 y(y—y / ) + x(b 2 —a' 2 )(x—x / ) = — ^ (cos /a + cos v), 
2b 2 -(x 2 +y 2 + a 2 ) = Rr, 
x(b 2 -a 2 )(y-y')-b 2 y(x-x r ) = -^ ( sin / i + sin ")> 
