212 
MR. W. H. BARLOW ON DIFFERENT MODES 
lens will be expressed by the number of times the surface of the central image 
is contained in the whole surface of the screen P, P" ; and this is true whether we 
consider the several images to be thrown in parallel lines, or condensed in a focus, 
or dispersed over a larger surface, for as the illuminated surface is contracted, the 
intensity is increased, and as it is extended, the intensity is diminished in the same 
proportion, so that under all circumstances the product of surface and intensity will 
be a constant quantity. Hence the illuminating power (abstracting from absorp- 
tion) will be increased by the reflector in the ratio of the surface of the lights to the 
surface of the end or section of the reflector. Or in other words, the area of the end 
of the reflector divided by the area of the light, will be a numerical measure of the 
illuminating power. 
This result is obtained by supposing the reflector to be composed of a number of 
small plane reflectors, each throwing the light in parallel lines, and each image 
therefore as having the same intensity as the direct light (screened as above) when 
viewed from the same distance ; but with a continuous curve surface, such as a para- 
bolic reflector, we must consider the divergency of the emanating ray at the point 
where it falls on the reflector, which will vary inversely as the square of the distance 
of that point from the centre of the light, or directly as the square of the sine of half 
the angle which the light subtends from that point, and therefore as the versed sine 
of half the same angle ; and the sum of all these must be compared with the area of 
the reflector, that is of its section or end, which varies also as the versed sine of half 
the angle which its extreme edge subtends at the light. 
In order, therefore, to compute the increase of illuminating power due to a para- 
bolic reflector, according to this principle, we must find a mean focal distance , that 
is, a distance (from which to estimate the constant angle subtended by the light) 
that shall be equivalent to the several variable distances. 
Let A D B be a parabolic reflector and C its focus, then will D C be the minimum 
and A C the maximum focal distance. Now if the light at C emanated from a point, 
all the rays intercepted by the surface A D B would be projected forward in parallel 
lines and cover the plane surface GH = AB at whatever distance it might be placed 
