OF ILLUMINATING LIGHT HOUSES. 
213 
from the reflector, and the light at G, K, L would be that due to the distances A C, 
I C, D C respectively : if then a segment of a sphere ni on be described intercepting 
the same number of rays as ADB, and whose surface is equal to the area of A B or 
H G, we shall have the same quantity of light equally distributed over the same 
surface ; hence the radius of the segment mon will be the mean focal distance with 
which all the light may be conceived to leave the reflector. 
Describe the circle A E B ; then, because A D B is a parabola, and A E B a circle 
described about it with the radius C A, and because C A = D F -f- D C, the height 
D F of the parabola = ^ the height E F of the segment A E B. 
ButEF = EC + CF = AC + CF, therefore D F = , 
and D C the minimum focal distance 
= D F — C F = 
A C - C F 
2 
Let A C = r, 
T — Tl 
C F = h, then r = maximum focal distance, and — — = minimum focal distance, 
(2 r X 3' 1416) (r + h)= surface of segment A F B ; and 4 (r 2 — h 2 ) . 7854 = area 
of AB or GH. 
Let x — radius of segment m o n : now the surface of the segment A E B is to the 
surface of the segment mon as r 2 to x 2 , and the area of the end A B is equal to the 
surface of the segment m o n, therefore 
(2 r X 3' 1461) (r -j- h) : 4 (r 2 — h 2 ) *7854 : :r 2 : x 2 
or 
2 r (r -|- h) x 2 — (r 2 - 
- h 2 ) r 2 
whence 
— (^ 2 ) r 2_ 
2 r (r -f h) 
C;") 
or 
x = \/ r ( Vb 
Y — 
But r = maximum focal distance and — — minimum focal distance. Therefore x, 
the mean focal distance, is a mean proportional between the maximum and minimum 
focal distances. Let therefore A represent the angle subtended by the reflector from 
the centre of the light, and a — the angle subtended by the light from the reflector 
at the mean focal distance, then 
versed sine | A 
versed sine I a ' 
will be the amount of illuminating power obtained by the reflector, that of the lamp 
being 1. 
This result differs in its numerical value very little from the former, viz. the area 
of the reflector divided by the area of the light. Thus, for example, let a reflector 
whose maximum focal distance is twelve inches, and minimum three inches, be iilu- 
2 f 2 
