If the candle-power had been uniform throughout the lower hemisphere at a value equal to the actual maximum of 100 the total number of lumens would have been 628, or just twice the actual value. Similarly, if the uniform candle-power of 100 has been active throughout both the upper and the lower hemisphere, the lumens output from the source would have totalled 1256, or four times the actual value determined above by slide-rule computation. The mathematically exact result would be,

The exact ratio between the total lumens produced by the lighting source having the candle-power distribution indicated in Fig. 2, and the lumens that would be produced by a source giving uniform candle-power in all directions equal to the maximum in Fig. 2, is 14. Obviously this ratio, which is called the "spherical reduction factor," in any practical case depends upon the shape of the candle-power distribution curve, becoming indefinitely small in the case of a concentrated beam and reaching a maximum of 1.0 in the case of a source of uniform candle-power such as a spherical surface source.

It may be well at this point to call attention to the fact that the "mean spherical candle-power" of a surface source of any shape whatsoever is equal to one-fourth of the product of the effective radiating area by the maximum candle-power of an (infinitesimal) unit area of the source, provided only that each infinitesimal area radiates in space according to the cosine law of space-distribution and all infinitesimal areas have the same maximum value of candlepower. The total effective candle-power in any chosen direction observed at any chosen position from such a source is equal to the product of the candle-power per unit area by the "projected area" of the source as viewed from the direction (and exact position) chosen. These facts will be discussed in greater detail later in connection with the subjects of "brightness" "output" and appearance."

On account of the fact that such curves as those shown in Fig. 2 are often loosely referred to as "light-distribution" curves, rather than "candle-power-distribution" curves, certain misconceptions have been produced in the minds of persons not familiar with the exact physical significance of the geometrical representation of the photometric relations.

In order to lay proper emphasis on the distinction that must be

made between "light distribution" and "candle-power distribution," a comparison will be made with the actual distribution of light in each vertical zone (as accurately shown by the Rousseau diagram of Fig. 3) and the distribution of light which would exist if the curve of Fig. 2 were in reality a "light distribution" rather than a "candlepower distribution" curve. This curve is reproduced in Fig. 4, where it is treated as representing "light distribution," and on the basis of this interpretation the Rousseau diagram of Fig. 5, has been constructed by the methods already explained. A comparison of the incorrect diagram of Fig. 5, with the correct diagram of Fig. 3 will serve to show the inaccuracy in treating a "candle-power distribution" curve as a "light distribution" curve.

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Figs. 4 and 5.-Space distribution of light from an assumed source and corresponding flux summation diagram.

CANDLE POWER DISTRIBUTION FROM CYLINDRICAL AND

SPHERICAL SURFACE SOURCES

In Fig. 6, the smaller double circles show the space distribution. of candle power around an infinitesimal cylindrical surface source having a vertical axis. In Fig. 7, the elliptical area is the Rousseau diagram showing the light flux produced over various zones of the sphere surrounding the light source, as explained above.

In Fig. 6, the large central circle shows the candle-power distribution around a spherical surface source; the corresponding Rousseau diagram is represented by the rectangular area in Fig. 7. The separate curves of Fig. 6 have been so drawn that the rectangular area of Fig. 7 is equal to the elliptical area of the same figure. That

is, the light output from the cylindrical surface has been made equal to the light output from the spherical surface source.

It will be recalled, from well-known trigonometrical and geometrical relations, that the area of an ellipse is equal to π/4 times the product of the major and minor axes, whereas that of a rectangle is equal to the product of the major and minor sides. It follows therefore that the minor side of the rectangle in Fig. 7 is equal to

Figs. 6 and 7.-Space distribution of candle-power from infinitesimal cylindrical and spherical sources and corresponding flux summation diagrams.

/4 times the minor axis of the ellipse, and hence the maximum (uniform) candle-power of the spherical surface source is equal to π/4 times the maximum (horizontal) candle-power of the cylindrical surface source in Fig. 6. That is to say, the "spherical reduction factor" of a cylindrical surface source is equal to /4= 0.7854. This is the value usually assigned to a so-called "line-source," which has no existence in reality, its nearest approach in practice

being the cylindrical surface of a lamp filament having an inappreciable diameter.

SPACE REPRESENTATION OF CANDLE-POWER DISTRIBUTION

By means of models representing solids of revolution of the candle-power curves about the axis of reference one can obtain a better idea of the real significance of the space distribution of the candle-power than can be obtained from the flat candle-power curve which must in any event be interpreted as showing merely a crosssectional view of such a space-model. In interpreting a candlepower distribution model care must be exercised in giving significance to the quantities represented. Special emphasis must be placed on the fact that neither the volumetric content of the model nor the superficial area has any immediate relation to the flux of light from the source giving the candle-power indicated by the model. A striking illustration of this fact is afforded by a comparison of the centrally located candle-power circle in Fig. 6 with the completely displaced candle-power circle in Fig. 2.

As already shown by means of the Rousseau diagrams of Fig. 7 and Fig. 3, the flux produced by the source giving the circular candlepower curve of Fig. 6 is exactly equal to that produced by the source giving the circular candle-power curve of Fig. 2, and hence the solid of revolution of Fig. 6 represents exactly the same amount of flux as does the solid of revolution of Fig. 2.

The diameter of the circle in Fig. 2 is exactly twice as great as that in Fig. 6; the superficial area of the solid of revolution of Fig. 2 is four times that of Fig. 6, and its volumetric content is eight times as large.

A certain percentage of the volumetric content or superficial area of any chosen solid of revolution represents the same percentage of the total flux of light from the source only in the limiting cases of uniform candle-power in all directions as shown by the centrally located circle of Fig. 6 or of a section of the sphere cut vertically throughout the whole depth.

From the two illustrations chosen above, it will be observed that even when the scale of candle-power is defined, the total flux represented by a given solid of revolution is known only when the exact location of the light source within the sphere is known. With the source at the center, the sphere represents the maximum of light flux; when the source is at the surface (as in Fig. 2) the light flux

has only one-half of the maximum value, all other quantities, dimensions and scales remaining the same.

SPHERICAL SURFACE: THE SO-CALLED "POINT "-SOURCE

For many purposes it has been found convenient to refer to a source of light as though it were a "point" (that is, without dimensions) and by certain mathematical transformations certain equations applicable exclusively to surface sources have been treated as though they related to true point-sources. When dealing with illumination effects at a distance, no measurable errors are involved in such assumptions and transformations, but when one attempts to define the "brightness" or appearance of the source to the eye on the basis of an assumed point-source, the assumptions are found to be at conflict with the most significant physical fact, which is that the brightness is a function of the area, whereas points (even an infinite number of them) are devoid of dimensions or area.

By treating the so-called "point-source," not as a true point but as an infinitesimal surface having all of the physical characteristics of a surface source the mathematical difficulties can be overcome, but by far the simplest and most satisfactory method is to treat the source initially, finally and all the time, as a surface source having true surface source characteristics.

=

Consider, therefore, a spherical surface source of unit radius (1) cm.) emitting 100 candle-power uniformly in all directions. The total output from the source will be 4 X 100 = 1257 lumens. The superficial area of the source is 4πr2 = 12.57 sq. cm., and hence the output is equal to 100 lumens per square centimeter. At any appreciable distance from the source the "projected area of the source viewed from this distance is equal to πr2 3.14 sq. cm. and hence the "apparent candle-power per unit of projected area" is 100 ÷ 3.14 = 31.9 a value which in the past has been called "brightness," but no name has been adopted for designating the unit. For the unit quantity "apparent output" from the source expressed in apparent lumens per sq. cm." the term "lambert" has been adopted. This term is applicable equally to the "brightness" (or appearance to the eye) of any surface whether radiating, transmitting, or reflecting, and whether or not it acts as a perfectly diffusing surface, but the unit is defined by, and receives its magnitude from, the appearance to the eye of "a perfectly diffusing surface radiating or reflecting one lumen per sq. cm.”