r = R cos θ.
Hence the area of the zone swept out by PQ is equal to
2πR cos θ ds = 2πR2 cos θdθ
in the limit, and the total quantity of light falling on the zone is equal to the product of the mean intensity or candle-power I in the direction AP and the area of the zone, and therefore to
2πIR2 cos θdθ.
Let I0 stand for the mean spherical candle-power, that is, let I0 be defined by the equation
4πR2I0 = 2πRΣ(Iab)
where Σ(Iab) is the sum of all the light actually falling on the sphere surface, then
| I0 = | 1 | Σ(Iab) = | Σ(Iab) | Imax |
| 2R | 2RImax |
where Imax stands for the maximum candle-power of the arc. If, then, we set off at b a line bH perpendicular to DE and in length proportional to the candle-power of the arc in the direction AP, and carry out the same construction for a number of different observed candle-power readings at known angles above and below the horizon, the summits of all ordinates such as bH will define a curve DHE. The mean spherical candle-power of the arc is equal to the product of the maximum candle-power (Imax), and a fraction equal to the ratio of the area included by the curve DHE to its circumscribing rectangle DFGE. The area of the curve DHE multiplied by 2π/R gives us the total flux of light from the arc.