is called the equator, we might consider the sphere divided
up by lines of latitude into zones, each of which would be
differently illuminated. The total quantity of light or the total
illumination of each zone is the product of the area of the zone
and the intensity of the light falling on the zone measured
in candle-power. We might regard the sphere as uniformly
illuminated with an intensity of light such that the product of
this intensity and the total surface of the sphere was numerically
equal to the surface integral obtained by summing up the products
of the areas of all the elementary zones and the intensity of the
light falling on each. This mean intensity is called the mean
spherical candle-power of the arc. If the distribution of the
illuminating power is known and given by an illumination
curve, the mean spherical candle-power can be at once deduced
(La Lumière électrique, 1890, 37, p. 415).
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| Fig. 9. |
Let BMC (fig. 9) be a semicircle which by revolution round the diameter BC sweeps out a sphere. Let an arc be situated at A, and let the element of the circumference PQ = ds sweep out a zone of the sphere. Let the intensity of light falling on this zone be I. Then if θ ≈ the angle MAP and dθ the incremental angle PAQ, and if R is the radius of the sphere, we have
also, if we project the element PQ on the line DE we have
| ab = | ds cos θ, |
| ∴ ab = | R cos θdθ |
| and Iab = | IR cos θdθ. |
Let r denote the radius PT of the zone of the sphere, then
Hence the area of the zone swept out by PQ is equal to
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
Let I0 stand for the mean spherical candle-power, that is, let I0 be defined by the equation
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.
Owing to the inequality in the distribution of light from an electric arc, it is impossible to define the illuminating power by a single number in any other way than by stating the mean spherical candle-power. All such commonly used expressions as “an arc lamp of 2000 candle-power” are, therefore, perfectly meaningless.
 |
| Fig. 10. |
The photometry of arc lamps presents particular difficulties, owing to the great difference in quality between the light radiated by the arc and that given by any of the ordinarily used light standards. (For standards of light and photometers, see Photometer.) All photometry Photometry of arc.depends on the principle that if we illuminate two white surfaces respectively and exclusively by two separate sources of light, we can by moving the lights bring the two surfaces into such a condition that their illumination or brightness is the same without regard to any small colour difference. The quantitative measurement depends on the fact that the illumination produced upon a surface by a source of light is inversely as the square of the distance of the source. The trained eye is capable of making a comparison between two surfaces illuminated by different sources of light, and pronouncing upon their equality or otherwise in respect of brightness, apart from a certain colour difference; but for this to be done with accuracy the two illuminated surfaces, the brightness of which is to be compared, must be absolutely contiguous and not separated by any harsh line. The process of comparing the light from the arc directly with that of a candle or other similar flame standard is exceedingly difficult, owing to the much greater proportion and intensity of the violet rays in the arc. The most convenient practical working standard is an incandescent lamp run at a high temperature, that is, at an efficiency of about 212 watts per candle. If it has a sufficiently large bulb, and has been aged by being worked for some time previously, it will at a constant voltage preserve a constancy in illuminating power sufficiently long to make the necessary photometric comparisons, and it can itself be compared at intervals with another standard incandescent lamp, or with a flame standard such as a Harcourt pentane lamp.
In measuring the candle-power of arc lamps it is necessary to have some arrangement by which the brightness of the rays proceeding from the arc in different directions can be measured. For this purpose the lamp may be suspended from a support, and a radial arm arranged to carry three mirrors, so that in whatever position the arm may be placed, it gathers light proceeding at one particular angle above or below the horizon from the arc, and this light is reflected out finally in a constant horizontal direction. An easily-arranged experiment enables us to determine the constant loss of light by reflection at all the mirrors, since that reflection always takes place at 45°. The ray thrown out horizontally can then be compared with that from any standard source of light by means of a fixed photometer, and by sweeping round the radial arm the photometric or illuminating curve of the arc lamp can be obtained. From this we can at once determine the nature of the illumination which would be produced on a horizontal surface if the arc lamp were suspended at a given distance above it. Let A (fig. 10) be an arc lamp placed at a height h( = AB) above a horizontal plane. Let ACD be the illuminating power curve of the arc, and hence AC the candle-power in a direction AP. The illumination (I) or brightness on the horizontal plane at P is equal to
 |
| Fig. 11. |
Hence if the candle-power curve of the arc and its height above the surface are known, we can describe a curve BMN, whose ordinate PM will denote the brightness on the horizontal surface at any point P. It is easily seen that this ordinate must have a maximum value at some point. This brightness is best expressed in candle-feet, taking the unit of illumination to be that given by a standard candle on a white surface at a distance of 1 ft. If any number of arc lamps are placed above a horizontal plane, the brightness at any point can be calculated by adding together the illuminations due to each respectively.
The process of delineating the photometric or polar curve of intensity for an arc lamp is somewhat tedious, but the curve has the advantage of showing exactly the distribution of light in different directions. When only the mean spherical or mean hemispherical candle-power is required the process can be shortened by employing an integrating photometer such as that of C. P. Matthews (Trans. Amer. Inst. Elec. Eng., 1903, 19, p. 1465), or the lumen-meter of A. E. Blondel which enables us to determine at one observation the total flux of light from the arc and therefore the mean spherical candle-power per watt.
 |
| Fig. 12. |
In the use of arc lamps for street and public lighting, the question of the distribution of light on the horizontal surface is all-important. In order that street surfaces may be well lighted, the minimum illumination should not fall below 0.1 candle-foot, and in general, in well-lighted Street arc lighting. streets, the maximum illumination will be 1 candle-foot and upwards. By means of an illumination photometer, such as that of W. H. Preece and A. P. Trotter, it is easy to measure the illumination in candle-feet at any point in a street surface, and to plot out a number of contour lines of equal illumination. Experience has shown that to obtain satisfactory results the lamps must be placed on a high mast 20 or 25 ft. above the roadway surface. These posts are now generally made of cast iron in various ornamental forms (fig. 11), the necessary conductors for conveying the current up to the lamp being taken
