LECTURE X. 
LIGHT FLUX AND DISTRIBUTION. 

86. The light flux of an illuminant is its total radiation 
power, in physiological measure. It therefore is the useful 
output of the illuminant, and the efficiency of an illuminant 
thus is the ratio of the total light flux divided by the power 
input. 

In general, the distribution of the light flux throughout space 
is not uniform, but the light-flux density is different in different 
directions from an illuminant. 

Unit light-flux density is the light-flux density which gives 
the physiological effect of one candle at unit distance. The 
unit of light flux, or the lumen, is the light flux passing through 
unit surface at unit light-flux density. The unit of light inten- 
sity, or one candle, thus gives, if the light-flux distribution is 
uniform in all directions, unit flux density at unit distance from 
the radiator, and thus gives a total flux of light of 4 it units, or 
4 it lumens (since the area at unit distance from a point is the 
surface of a sphere, or 4 it). 

The unit of light intensity, or the candle power thus given, 
with a radiator of uniform light-flux distribution, 4 x lumens of 
light flux, and inversely, a radiator which gives 4 it lumens 
of light flux, gives an intensity of one candle, if the intensity is 
uniform in all directions, and, if the distribution of the intensity 
is not uniform, the average or mean spherical intensity of the 
radiator is one candle. Thus one mean spherical candle rep- 
resents 4 it lumens of light flux, and very frequently the mean 
spherical candle is used as representing the light flux: the light 
flux is 4 TT times the mean spherical intensity, and the mean 
spherical intensity is the total light flux divided by 4 it, regard- 
less whether the light flux is uniformly distributed or not. 

The total light flux of an illuminant is derived by the sum- 
mation or integration of the intensities, that is, the flux den- 
sities at unit distance, in all directions from the radiator. 

186 


LIGHT FLUX AND DISTRIBUTION. 187 

The distribution of light flux or of intensity is never uniform, 
and the investigation of intensity distribution of the light flux 
thus necessary. 

The distribution of the light intensity of an illuminant de- 
pends upon the shape of the radiator and upon the objects 
surrounding it; that is, the distribution of the light flux issuing 
from the radiator depends on the shape of the radiator, but 
is more or less modified by shadows cast by surrounding 
objects, by refraction, diffraction, diffusion in surrounding 
objects, etc. 

The most common forms of radiators are the circular plane, 
the straight line, that is, the cylinder, the circular line or circular 
cylinder and combinations thereof. 

87. Very frequently the intensity distribution of an illumi- 
nant is symmetrical, or approximately symmetrical, around an 
axis. This, for instance, is the case with the arc lamp, the 
incandescent lamp, most flames, etc. If the distribution is 
perfectly symmetrical around an axis, the distribution in space 
is characterized by that in one meridian, that is, one plane pass- 
ing through the axis. If the distribution is not symmetrical 
around the axis, usually the space distribution is characterized 
by the distribution curves in two meridians at right angles to 
each other, the meridian of maximum and that of minimum 
intensity, and the distribution in the equatorial plane, that is, 
the plane at right angles to the axis. 

Distribution curves are best represented in polar coordinates, 
and the angle <f> counted from the axis towards the equator 
(that is, complementary to the " latitude" in geography). 

As most illuminants are used with their symmetry axis in 
vertical direction, and the downward light is usually of greater 
importance, it is convenient in plotting distribution curves to 
choose the symmetry axis as vertical, and count the angle $ 
from the downward vertical towards the horizontal; that is, 
the downward beam would be given by (f> = 0, the horizontal 
beam by ^ = 90 deg., and the upward beam by <j> = 180 deg. 

The usual representation of the light-flux distribution in po- 
lar coordinates does not give a fair representation of the total 
light flux, or the mean spherical intensity of the light source, 
but on the contrary frequently is very misleading. When com- 
paring different polar curves of intensity distribution, it is 


188 RADIATION, LIGHT, AND ILLUMINATION. 

impossible to avoid the impression of the area of the curve as 
representative of the light flux. The area of the polar curve, 
however, has no direct relation whatever to the total light flux, 
that is, to the output of the illuminant, since the area depends 
upon the square of the radii, and the light flux directly upon 
the radii of the curve. Thus an illuminant of twice the inten- 
sity, but the same flux distribution, gives a polar curve of four 
times the area, and the latter gives the impression of a source 
of light far more than twice as great as the former. 

The meridian curves of intensity distribution are still more 
misleading: the different angles of the curve correspond to 
very different amounts of light flux: the horizontal intensity 
(<£ = 90 deg.) covers a zone of 2 rn circumference, while the 
intensity in any other direction (f> covers a zone of 2 rn sin <j> 
circumference; that is, an area which is the smaller, the nearer 
(f> is to 0 or 180 deg. ; the terminal intensity, upward or down- 
ward, finally covers a point only, that is, gives no light flux. 
As the result hereof, an illuminant giving maximum intensity 
in the downward direction, and low intensity in the horizontal, 
gives a much larger area of the polar curve than an illuminant 
of the same or even a greater total light flux which has its 
maximum intensity in the horizontal. Comparing, therefore, 
illuminants of different distribution curves, it is practically 
impossible not to be misled by the area of the polar curve, and 
thus to overestimate the illuminant having maximum downward 
intensity, and underestimate the illuminant having maximum 
horizontal intensity. 

The misleading nature of the polar curves of intensity dis- 
tribution in the meridian is illustrated by the curves in Figs. 
64 and 99: the three curves of Fig. 64 give the same total 
light flux; that is, the same useful output; but 2 looks vastly 
greater than 1 or 3, and 3 especially looks very small. Curves 
0, 1, 2, 3, 4 in Fig. 99 give the same total light flux, and curve 
5 gives only one tenth the light flux. To the eye, however, the 
curve 4 gives the impression of a far more powerful illuminant 
than the curve 1, and curve 5 appears practically equal, if not 
larger than 1, while in reality it represents only one tenth the 
light output of 1. 

88. In an illuminant in which the distribution of intensity 
is symmetrical around an axis, and thus can be represented 


LIGHT FLUX AND DISTRIBUTION 


189 


by one meridian curve, the total light flux is calculated thus: 
Let / = intensity at angle <f> 

(counting the angle <j> from one pole over the equator to the 
other pole). 

This intensity covers a zone of the 
sphere of unit radius of width d(j> and 
angle <£,that is, a zone of radius (Fig. 62) 
r = sin<£; thus surface 

dA = 2 n sin 


and the light flux in this zone therefore 
is: 


FIG. 62. 


= 27r/sin<M^ 
hence, the total light flux : 

<£ = 2 TT / /sin (f>d(/>. 
Jo 


CD 


(2) 


The light flux in the space from the downward direction </> = 0 
to the angle <£ = fa against the vertical or symmetry axis, then 
is 


fc1 = 2 TT / sin <t>dfa (3) 

*/0 

and the light flux in a zone between the angles (j)1 and fa is 

(4) 


I. DISTRIBUTION CURVES OF RADIATION. 

(1) Point, or Sphere, of Uniform Brilliancy. 

In this case, the intensity distribution is uniform, and thus, if 

/ = intensity of light, in candles, 
<£= 4 nl = light flux, in lumens; (5) 

or, inversely: 

<I> 
/ = -. (6) 

The brilliancy of a radiator is the light-flux density at its sur- 
face. Thus, with a luminous point of intensity /, the brilliancy 


190 


RADIATION, LIGHT, AND ILLUMINATION. 


would be infinite; with a luminous sphere of uniform intensity 
distribution, and of radius r, the brilliancy is 

* 7 ' (7) 


B = 


,2 ' 


hence, inversely proportional to the square of the radius of the 
spherical radiator. 

(2) Circular Plane of Uniform Brilliancy. 
89. Such radiators are, approximately, the incandescent tip 
of the carbons in the (non-luminous) electric carbon arc, or 
the luminous spot in the lime cylinder 
of the lime light (hydro-oxygen flame), 
etc. 

Choosing the circular luminous plane 
as horizontal direction, the intensity 
distribution is symmetrical around the 
vertical, the vertical direction thus can 
be chosen as axis, and the angle <£ 
counted from the vertical upward. ' 

The intensity is a maximum 70, ver- 
FlG- 63- tically downward, for <£ == 0. 

In any other direction, under angle (f> against the vertical 
(Fig. 63), the intensity is 

/ = /0 cos <£, (8) 

and is zero for 

<£ = 90 deg. 

The light flux issuing from the radiator below angle <f> is, by 
(3): 


hence, by (8) : 


rsin (j) cos (f)d<f> 


-I. /cos 2* 

& I I 


-/„{! -cos 24,}, 


(9) 


LIGHT FLUX AND DISTRIBUTION. 191 

and the total light flux, from (f> = 0 to <j> = 90 deg. = — , thus is : 

ft 


or 


The brilliancy of the source of light is the total light flux 
divided by the luminous area; or, 

* 


and, if r = radius of the luminous circle, 

A = ;rr2, 
and 


-?'' 02) 

or, 

70 = r2£; (13) 

that is, the same as in class (1). 

Comparing (11) with (5), it thus follows that the total light flux 
of such a radiator, for the same maximum intensity, is only one 
quarter that of a radiator giving uniform intensity distribution 
throughout space, or inversely, with such a downward distribution 
of light, the maximum intensity is four times as great as it would 
be with the same total light flux uniformly distributed through 
space. 

The flux distribution is a circle having its diameter from the 
source of light downward. It is shown as 2 in Fig. 64, and 
the concentric circle giving uniform intensity distribution of 
the same total light flux is shown as 1. 

(3) Hollow Circular Surface. 

Such a radiator, for instance, is approximately the crater of the 
positive carbon of the arc lamp. 

As with such a radiator, as shown in section in Fig. 65, the 
projection of the luminous area in any direction <f> is the same 


192 RADIATION, LIGHT, AND ILLUMINATION. 


FIG. 64. 


FIG. 65. 


FIG. 66. 


LIGHT FLUX AND DISTRIBUTION. 193 

as with the plane circular radiator (2), the same equations 
apply. 

(4) Rounded Circular Surface. 

Such, for instance, is approximately the incandescent carbon 
tip of the arc-lamp electrodes, when using carbons of sufficiently 
small size, so that the entire tip becomes heated. 

Assuming, in Fig. 66, the radiator as a segment of a sphere, and 
let 2 aj = the angle subtending this segment, rl the radius of this 
sphere. 

For all directions <£, up to the angle co below the horizontal : 

0< 0 <£-w ; 

the projection of the spherical segment in Fig. 66 is the same as 
that of a plane circle, and thus the intensity is given in class (1), 
as: 

i = 70 cos (/>. 

In the direction, -- - co <<£<-, however, the intensity is 

A 2i 

greater, by the amount of light radiated by the projection Dyx, 
and, in the horizontal direction, the intensity does not vanish, 
but corresponds to the horizontal projection of the luminous 
segment. 
Above the horizontal, light still issues in the direction, 


from the segment Buv, and only for - + co < <j> does the light 

Zi 

cease. 

If r2 = radius of carbon, the radius of the luminous segment is 


sin co' 


the height of the segment is 

h = rl (1 — cos co) 
r2 (1 — cos co) 


sin co 


194 RADIATION, LIGHT, AND ILLUMINATION, 
hence the surface of the segment, or the luminous area, is 
A2 = 2 rjin 

2 r227r (1 — cos w) 


2r227r X 2 sin2 


• 2 2 

4 sin2 - cos - 
Z 2 


COS 2 


(14) 


Thus, if the luminous area is the same as in the plane circle 
class (2), it must be: 

TV2 7T 


r2 = rcos~; (15) 

and, if the brilliancy B is the same, the maximum intensity for 
<t> = Ois 


cos2 ; (16) 

that is, the rounding off of the circular radiator, at constant bril- 
liancy and constant luminous surface, decreases the maximum 

intensity 70 by the factor cos2-, but increases the intensity within 

the angle from co below to a> above the horizontal direc- 
tion. 

In Fig. 67 are plotted the distribution curves, for the same 
brilliancy and the same area of the radiator, for a plane circular 
radiator, as 1 ; a rounded circular radiator of angle co = 30 deg. 
as 2, and a rounded circular radiator of angle a> = 60 deg., 


LIGHT FLUX AND DISTRIBUTION. 


195 


as 3. As seen, with increasing rounding, gradually more and 
more light flux is shifted from the vertical into the horizontal 
direction. 


FIG. 67. 


Straight Line or Cylindrical Radiator. 

90. Such radiators are represented approximately by the lum- 
inous arcs with vertical electrodes, by the mercury-arc tube, 
by straight sections of 
incandescent-lamp fila- 
ments, etc. 

The intensity distribu- 
tion is symmetrical with 
the radiator as axis. 

The intensity is a max- 
imum 70 at right angles 
1,1 T i • FIG. 68. 

to the radiator, or in 

horizontal direction, <j> = 90 deg., when choosing the radiator as 
vertical axis. At angle <£, the intensity is, Fig. 68, 

I = /o sin 0, (17) 

and is zero for </> = 0 and (f> = 180 deg., or in the vertical. 


196 RADIATION, LIGHT, AND ILLUMINATION. 

The light flux within angle <f> from the vertical is, by (4), 


= 7r/0 f (l-co 

«^o 

hence, , A _ / . sin 2 


and the total light flux for (f> = K is 

* = »v.; (19) 

or, inversely: ^_ ^ ? : (2Q) 

and the radiating surface is 

A = ^ (21) 

where Z is the length; w the diameter of radiator. The bril- 
liancy, therefore, is ^ 


(22) 


may be called the linear maximum intensity, or, maximum inten- 
sity per unit length. 

Most of the light 'of a linear vertical radiator issues near the 
horizontal, very little in downward and upward direction. Put- 
ting <J>0 = J $, gives the angle <f>, which bisects the light flux : 

sin 2 (f> n 
* — =4' 

and herefrom, by approximation, $ = 66 deg.; that is, half the 
light flux issues within the narrow zone from 24 deg. below to 


LIGHT FLUX AND DISTRIBUTION. 


197 


24 deg. above the horizontal, or in the space between a and a' in 
Fig. 68. 

It is interesting to compare the three radiators, (1), (2), and (5), 
on the basis of equal maximum intensity, and on the basis of 
equal light flux, thus : 


Light flux <£, at equal maximun 
intensity /0 

Maximum intensity 70, at equal 
light flux $ 


Uniform. 
47T/0 

4 

4^ 
1 


Circle. Cylinder. 


7T= 3.14 


1=1.27 

7T 


As seen, at the same maximum intensity, the cylinder gives 
nearly as much light flux as given by uniform distribution, that 
is, its deficiency in intensity in the polar regions represents very 
little light flux. The circular plane, however, gives only one 
quarter as much light flux as uniform distribution. 

With the same horizontal intensity of a cylindrical radiator, 
as the vertical intensity of a circular plane, the former gives 
TT = 3J4 times the flux of light. 

In Fig. 64 the three distribution curves are shown for the same 
total flux of light: curve 1 for uniform intensity, 2 for a plane 
circle, and 3 for a straight cylinder as radiator. 

(6) Circular Line or Cylinder. 

In the spirals, loops or ovals of in- 
candescent-lamp filaments, circular 
radiators, or sections thereof, are met. 

Let r = radius of the circular radi- 
ator, w = diameter of the radiator 
cylinder, shown in section in Fig. 69. 

The intensity is a maximum in the 
direction at right angle to the plane 
of the circle. 

The projection of the radiator in this direction of maximum 
intensity, $ = 0, has the length: 2 TIT; and if, by (24) : 

wB 


69- 


maximum linear intensity, 


198 RADIATION, LIGHT, AND ILLUMINATION. 

where B = brilliancy, 

it is, IQ=27rrIQ'=2rwB. (25) 

This is in the direction in which the projection of the radiator 
is a circle of radius r, and thus circumference 2 TTT. 

In any other direction </>, the projection of the radiator is an 
ellipse, with r and r cos (f> as half axes, as seen from Fig. 69. 

If I = the circumference of this ellipse, the intensity in the 
direction <j> bears to the maximum intensity 70 the same ratio as 
the circumference of the ellipse to that of the circle; that is, 

I-It^—Ifl. (26) 

2 nr 

The circumference of an ellipse with the half axes a and c is 

I = (a + c) n (1 + q), 

I/a - c\2 1 /a - c\4 1 /a - c\8 W 

4 \a + c) + 64 Va + c) + 256\a + c) + ' ' 


where 


The ratio of the circumference of the ellipse to its maximum 
diameter, y = — , is given in Table I, and plotted in Fig. 70, 

w T 

with the ratio of the half axes, that is, cos <p, as abscissas, and, in 
Fig. 71, with angle <f> as abscissas. 

TABLE I. — CIRCUMFERENCE OF ELLIPSE. 


c 

- — COS d>. 

a 

V = Tr- 

f 

V=ir- 

1.0 

1.571- s 

0 

1.571 = 5 

2 

2 

0.9 

.495 

10 

1.560 

0.8 

.418 

20 

1.525 

0.7 

.345 

30 

1.470 

0.6 

.278 

40 

1.390 

0.5 

.210 

45 

1.350 

0.4 

.150 

50 

1.305 

0.3 

.110 

60 

1.220 

0.2 

.055 

70 

1.120 

0.1 

.025 

80 

1.045 

0 

.000 

90 

1.000 

LIGHT FLUX AND DISTRIBUTION. 


199 


\ 

y - 

1.5 
1.4 
1.3 
1.2 
1.1 
1.0 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 

\ 

X 

N 

X 

X. 

X 

^ 

\ 

^ 

C( 

£ 

minimum p, 
mayimu/Ti 

imet 

erof 

Ellii 

sis 

0 

I 0 

,i. 

4 0*5 0 

6 0 

7 0 

8 0 

9 

-^v 

1.5 
1.4 
1.3 
1.2 

1.1 
1.0 
0.0 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
'0.1 

^ 

\ 

\ 

\ 

^ 

x 

^ 

V, 

1 

2 

Angle^Deglrees 
0 3.0 40 1?0 6 

0 *i 

tt f 

> 

FIG. 70. 


FIG. 71. 


In Fig. 72 is plotted the intensity distribution in the meridian 


of such a circular radiator. 


This shows a maximum 70 in the 


vertical, and a minimum ^= — 70 

7T 

in the horizontal. 

Theoretically, exactly in the 
horizontal, <f> = 90 deg., the in- 
tensity should be ~, as one half 
ft 

of the circle shades the other 
half. In most cases of such circular 
radiators, sections of incandescent- 
lamp filaments, w is so small com- 
pared with r, that it is practically 
impossible to have the radiator 
perfectly in one plane, as would be 
required for one half to shade the 
other half. 

(7) Single-Loop Filament. 


FIG. 72. 


200 


RADIATION, LIGHT, AND ILLUMINATION. 


92. As an illustration of the use of the distribution curves 
of different typical forms of radiators, the distribution curves 
of a single-loop incandescent-lamp filament may be calcu- 
lated. 

Such a filament consists of two straight sides, joined by a half 
circle, as shown in Fig. 73. 

The distribution of in- 
tensity is not symmetrical 
around any axis, but ap- 
proximately so around the 
axis Z in Fig. 73. 

The meridian of maxi- 
mum intensity is the plane 
YZ, at right angles to the 
plane of the filament ; the 
meridian of minimum in- 
tensity is the plane of the 
filament, XZ, and the 
least variation of intensity 
occurs in the equatorial 
plane XY. The distribu- 
tion curves in all three of 
.these planes are required. 
Assuming the straight 
sides as of a length equal 
to twice the diameter of 
the loop, or of length 4 r, 
where r = radius of the half circle. 

As it is impossible to produce and maintain such a filament 
perfectly in one plane, we assume, as average deviation of the two 
straight sides A and B of Fig. 73 from the vertical, an angle of 
10 deg. 

The intensity distribution of the straight sides A and B in any 
meridian plane thus is that of a straight radiator, (5), at an 
angle of 10 deg. against the vertical. 

Let // = maximum intensity per unit length. Then the 
meridianal distribution of the sides A + B is : 


FIG. 73. 


7 = 4 r/ 


sn 


+ 10°) + sin (0 - 10°) } 


(28) 


Hereto in the meridian of maximum intensity is added the light 


LIGHT FLUX AND DISTRIBUTION. 201 

intensity produced by a half circle of radius r, (6); that is, 

'2 = £-' (29) 

where I is the circumference of the ellipse which projects the 
circle of radius r, under angle <£, and is given by Table I and 
Figs. 70 and 71. 


FIG. 74. 

In the meridian of minimum intensity, the light intensity 73 
produced by the projection of the half circle in its own plane, 
under angle (f>, is added to the intensity 7t. This projection is, 
by Fig. 73, 

c = r (1 + cos <£), (30) 

and thus 73 = c70' 

= rIQ' (1 + cos 0). (31) 

In the equatorial plane, the intensity, due to the straight sides 
A + B, is constant, and is that of a straight radiator under angle 
10 deg. from the direction of maximum intensity; hence is 

70 = 8 rI0' cos 10°. (32) 

To this is added the intensity produced by the half circle of 
radius r, that is, 72; hence, in the meridian of maximum intensity, 
7 = 7j + 72, Curve 1 of Fig. 74; in the meridian of minimum 


202 


RADIATION, LIGHT, AND ILLUMINATION. 


intensity, / = 7X 4- 73, Curve 2 of Fig. 74 ; and in the equator, 
7 = 70 + 72, Curve 3 of Fig. 74. 

(8) In Table II are recorded the intensity distribution of the 
different radiators discussed in the preceding paragraphs. 


TABLE II. 


Circular surface. 

Single-loop filament. 

* 

Rounded by 

Circular 

Meridian of 

Plane. 

30 deg. 

60 deg. 

Max. 

intensity. 

Min. 
intensity. 

Equator. 

0 

7.00 

6.50 

5.25 

3.14 

1.70 

1.70 

5.51 

10 

6.88 

6.40 

5.18 

3.12 

1.73 

1.68 

5.50 

20 

6.57 

6.12 

4.94 

3.05 

2.47 

2.35 

5.46 

30 

6.05 

5.63 

4.56 

2.94 

3.19 

2.97 

5.41 

40 

5.35 

4.98 

4.07 

2.78 

3.84 

3.53 

5.33 

50 

4.50 

4.19 

3.55 

2.61 

4.41 

4.02 

5.24 

60 

3.50 

3.25 

3.01 

2.44 

4.88 

4.41 

5.16 

70 

2.39 

2.31 

2.44 

2.24 

5.23 

4.70 

5.06 

80 

1.21 

1.47 

1.90 

2.09 

5.44 

4.88 

4.98 

90 

0 

0.75 

1.37 

2.00 

5.51 

4.94 

4.94 

100 

0.39 

1.00 

110 

0 09 

0 07 

120 

0 

0.04 

130 

0 02 

140 

0 005 

150 

0 

77. SHADOWS. 

93. The radiator of an illuminant can rarely be arranged so 
that no opaque bodies exist in its field of light flux and obstruct 
some light, that is, cast shadows. As the result of shadows, the 
distribution of intensity of the illuminant differs more or less 
from that of its radiator, and the total light flux is less. 

The most common form of shadow is the round shadow sym- 
metrical with the axis of the radiator, that is, the shadow of a 
circular plane concentric with and at right angles to the sym- 
metry axis of the illuminant. Such for instance are, approxi- 
mately, the shadows cast by the base of the incandescent lamp, 
by the top of the arc lamp, etc. Such also are the shadows of 


LIGHT FLUX AND DISTRIBUTION. 


203 


the electrodes in the arc lamp in that most common case where 
the electrodes are in line with each other. 

As an example may be considered the effect of a symmetrical 
circular shadow on the light flux and its distribution with a 
circular plane and with a straight line as radiator. 

(1) Circular Plane Opposite to Circular Plane of Radiator. 

Shadow of negative carbon in front of the positive carbon 
of the carbon arc. 

In Fig. 75, let 2 r be the diameter of a circular plane radiator 
(positive carbon) ; 2 rl 
the diameter of the 
plane, which casts a 
shadow (negative car- 
bon of the arc lamp); 
and I the distance be- 
tween the two. 

Assume 70 as the 
maximum intensity of 
the light flux issuing 
from the radiator AOB 
(which is in downward 
direction, hence com- 
pletely or partly intercepted by the circle Afl^BJ. Then, the 
intensity of the light flux from the radiator, in any direction $, 
is, according to reasoning under heading I, class (2) , 

7 = /0cos0. (1) 

In this direction <j>, the circle AlBl projects on the plane AB 
as a circle A2B2) with radius rv and the center 02 of this circle 
has from the center 0 of the radiator the distance 


FIG. 75. 


a = 00 


tan 


(2) 


If now the projected circle 02 overlaps with the radiator circle 
Ov the area S of overlap, shown shaded in Fig. 76, is cut out from 
the radiator by the shadow, and the light flux in the direction <j> 
thus reduced from that of the complete radiator surface, Trr2, 
to that of the radiator surface minus the shaded part S, that is, 
xr2-S, or in the proportion 


2* -S 


s 


r'x 


(3) 


204 RADIATION, LIGHT, AND ILLUMINATION. 

and the intensity of the remaining light flux, in the direction <f>, 
thus is I = I0q cos <£. (4) 

If the distance, a, between the circles 0 and 02 is greater than 
the sum of radii, I tan (j> > r + r17 the circles 0 and 02 do not 
overlap, and in that direction no shadow is cast. 

The light intensity thus is reduced by the shadow of the lower 
carbon only for those angles <j> which are smaller than the angle 

<t>i given by r + r 

~--A (5) 


In the direction in which <£ is smaller than the angle, 

r. — r 
tan ^2 = ~ — > (6) 

and the shadow 02 thus covers the entire radiator 0, no light 
issues, but the radiator is completely shaded. This can occur 
only if rt>r, and if this is the case, a circular area below the 
radiator receives no light. 

If fj= r, the intensity becomes zero only in the direction 
<j> = 0; and if 7\ < r, the light in the downward direction is merely 
reduced, but nowhere completely extinguished. 

The shaded area of the radiator consists of two segments, 
of the respective radii r and r1 : S = D + Dr 

Let 2 co = angle subtending segment D and 2 co^ = angle 
subtending segment Dv and denoting the width of the segments 
thus 

w = AC, 


and the total width of the shaded area is 

p = AB2 = w + w,. (7) 

From Fig. 76, 

a = 002 = OA + J~02 - AB2 

= r + r, - p; 
or, 

p = r + rt - a; 
hence, by (2), 

p = r + TI - tan 0. (8) 


LIGHT FLUX AND DISTRIBUTION. 
In A 02EO, 


205 


sin < 
sin 

sin ( 


r . 
- sm aj, 


and 


hence, 


Furthermore, 


S = D + D, 


and, by (3), 


(9) 

(10) 
(ID 

(12) 


(13) 

(14) 
(15) 

(16) 


For different values of w the 

values of to wt p D D, S q 

are calculated from equations (10) (11) (12) (13) (14) (15) (16) 
and then q plotted as function of p in Fig. 78. 


206 


RADIATION, LIGHT, AND ILLUMINATION. 


From equation (8) then follows, for every value of <£, the cor- 
responding value of p, herefrom the value of q and by (4) the 

value of /. 


fl 

1.0 

0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0 

*> 

N 

^ 

N 

^ 

\ 

^ 

\ 

> 

\\s 

^V 

in 

\ 

\ 

\I 

v 

I\ 

\ 

\ 

V 

\n 

P = 

5 

o 

1 0 

2 0 

3 0 

t 0 

5 0 

6 0 

7 0 

8 0 

^ 

FIG. 78. 


94. In Table III are given 
the values of p and q for the 
ratio of radii : 

^- = 2.0; 1.0; 0.7, 

corresponding to a shadow sec- 
tion equal to 4 times, 1 times, 
and 0.5 times the section of the 
radiator. 

These are plotted as curves 
I, II, III, in Fig. 78. 


TABLE III. 


P' 

r-l=2. 
r 

3-1. 

r 

TI 
r 

= 0.7. 

I. 

II. 

III. 

IV. 

0 

1.000 

1.000 

1.000 

0 96 

0.1 

0.962 

0.964 

0.968 

0.2 

0.885 

0.895 

0.910 

0.3 

0.785 

0.810 

0.835 

0.4 

0.675 

0.715 

0.748 

0.5 
0.6 
0 65 

0.560 
0.435 

0.615 
0.500 

0.660 
0.575 
0 532 

0.660 
0.569 
0 520 

0.7 
0 71 

0.308 

0.380 

0.502 
0 500 

0.477 

0.8 
0.85 

0.183 

0.255 

0.500 

0.386 
0 340 

0.9 

0.070 

0 127 

0 500 

0.95 

0.028 

0 063 

1.0 

0 

0 

0 500 

t 

If — < 1, the curve III represents the effective light-living 

area only up to the values of p, where wl = rv but beyond this 
value, at least in the application to the shadow cast by the 


LIGHT FLUX AND DISTRIBUTION. 207 

negative carbon of the arc lamp, the shaded area is not merely 
the circle O/, but also the area shown shaded in Fig. 77, 
which is shaded by the shadow cast by the sides of the lower 
electrodes. 

From the value p, which corresponds to wl = rv the area S 
then increases by 2^ (p-p)', hence, if S'= shaded area for 
p = p'j for any value of p>pv 


and q = l ;- ^j^. (17) 

XT nr2 

This is shown in Table III and in Fig. 78 as curve IV. 

Such curves of intensity of a plane circular radiator of radius 
r, shaded by a concentric circular shade of radius rl at distance I 
[corresponding to a diameter of positive carbon 2 r, of negative 
carbon 2 ri; and an arc length Z], are given in Figs. 79 to 82, 
and the numerical values given in Table IV. 

r I 

Fig. 79 gives the curves for - = 2, and the arc lengths, — = 

0.25; 0.5; 1.0; 2.0, as curves I, II, III, IV. Fig. 80 gives the 

r 2r 

curves for - = 1, and the arc lengths, —= 0.25; 0.5; 1.0; 2.0, 

r I r 

as curves I, II, III, IV. Fig. 81 gives the curves for - = 0.7, 

I r 

and the arc lengths, — - = 0.25; 0.5; 1.0; 2.0, as curves I, II, III, 

IV. In Fig. 82 are shown, for comparison, the intensity curves 
for 

i-2; 2-r=2'asL 
5-1; 1-1, as II. 

^ = 0.7; ~=0.5, as III. 

T £ T 

As seen from Fig. 82, a larger shade at greater distance, I, 
gives approximately the same light flux and a similar distribu- 
tion, but gives a much sharper edge of the shadow, while a smaller 


208 RADIATION, LIGHT, AND ILLUMINATION. 


V////A W/\ v////\ 


FIG. 81. 


FIG. 82. 


LIGHT FLUX AND DISTRIBUTION. 


209 


j 

0 

co 

i b- t^. o i^ o co o 10 10 o co »o o o »o o 
»-i o o <o t^ 10 co o t- •* o 10 o •<*< r^ o 

o »-• c<i M co ^ >o «o eo r^. oo oo oi o> o> d 

11 

S 3 

II 

0 

• ! !! t^.^t"o co»oo io»oo «oo»oo 
O-. •• cococo »o^co i-io>t^ eoo»oo 

. I . '. o i-i <M co •* »o «o <o t>. od o» o> d 

I C 

~l« 

IO 

<M ^ CO 00 OS 0 •-< S 0 

fl 

0 

O- OT-HCNCO-*»Ot>-oOOSO 

8 

10 

CO 

!! l!i ! 03COOOOOOOO 
. . . . . . CO IO Tj< CO CM O 

o 

o • • • • • . o o CM •«* co oo o 

T-H 

o 

t^OlO O5O5i— I t^C<lt-- i— 1 ^f t-» OC^C^ COt^-C3i 

(N 

CO »O CO t^ 00 OS 00 00 t^ t^ CO IO »O ^ CO CO rH O O 

H« 

0 

t^-*o t^coos -*oot^ ^nrt<t^ o«MTt< cot^os 

CO^lO tOCOCO t^t>»t^ t^CO>O »O^CO M i— I O O 

t>I 

d 

»0 
0 

t^^H-^ cOOSi-H CO-^iO i-^CO*O o<N-rt< COt^OS 

co-^Tfi T^T»<»O loioio io»oio iri^co co»-ioo 

Cl u 

IO 

co 

10 »o 
t^oso I-I<MCO CO<NT-I ^HOOS coioco cot^os 

i? 

o 

COCO'* -^Tt<Tl< rt<"*^ -«*<TJ<CO COCOCO <M'-"OO 

13 

o 

QOCO ^OCO CDOSCO b-CO-^ OCO(M OS-*t^ 
i— I CO COOOOO CO >— I CO O^ft^ OCMrfrl »Ot^QO 

a 

II 

CO 

OC^-* COt>-00 OOOOt^ t^-COO >OTt<CO CM»-IOO 

<o_ 
N 

-IS 

O 

COOS 00 t>- if* CO IO O t^ CO -* O CO CO OS •«*< t>- 
O--H CN(M(M OCO<M O^t^ OCN-* *Ot^OO 

.1 
"3 

*""* 

O>-i<N COTt<iO coiOIr^ t^eoiO lO-^CO CO»-iC>O 

2 
3 

a 

II 
*TI v. 

»o 
d 

COOO -^COOO t^OO OO<M iOCO(M OS-^t^ 

»0 0 CO ^H CO 1-H CO rH T*H CO t«- t- <M ^ »0 t>- 00 

OOi-< »-iCS|<M COCO-* -«^-^TJ< -«t*TjIcO COi-HOO 

1 

A 

iO 

co 

»O i— I t^»OCO COOCO t^COt^. COt^CO lO^t» 
CN >0 1^ 0 CO 10 00 0 (N Tt< »0 t>-t^t>. 10^.00 

J3 
3 

o 

OOO O«-l<-l r-i»-iCN COCMCO COCOCO CO^-iOO 

i 

0 

CO ^H -«** CO OS t^ >-irt<l^ 0 CO -«*< COt^OS 

ii 

CO 

O -O O CO Tj« CO t>- b, t^ CO >O U5 T*I CO CO ^i O O* 

„ 1 V- 

IN 

O 

«O »O 00 O OS CO o CO •** CO t- OS 

1—1 

O- O OT-ICO Tt*^iO lO-^CO CO«-«OO 

c4 

11 

•Tl t- 

. . . . . . . . - — 

»o 

CO 

i-i co co t^ o» 

0 

O • • ... ... ... OOO i-i ,-H O O 

•s 

88JS8Q 

•e- 

o 10 o »o o »o o m o to o »o o 10 o 10 o »o o 

ri t-HCOCO COCO-* ^IO«O COcOt^ t^OOOOOS 

210 RADIATION, LIGHT, AND ILLUMINATION. 

shade at shorter distance, III, gives a far broader half shadow, 
which extends even to the vertical direction. 

This is illustrated by the distribution of the open arc and the 
enclosed arc in clear globes — that is, without means of diffrac- 
tion or diffusion. In the enclosed arc the distance between the 

electrodes, I, is made larger, since the ratio of radii — is greater, 

as due to the smaller current the diameter of the radiator, 2 r, 
is smaller than in the open arc. The enclosed arc has a much 
sharper edge of the shadow, that is, narrower half shadow, than 
the open arc, thus requiring means of diffusion of the light even 
more than the open arc. 

Where the shade which casts the shadow is rounded, the dis- 
tribution curve is somewhat modified by similar considerations, 
as have been discussed under headings I, class (4). This is fre- 
quently the case where the shadow is cast by the electrodes of an 
arc, and especially so in the carbon arc, in which the negative 
electrode — which casts the shadow — is more or less rounded 
by combustion. 

(2) Circular Plane Concentric with the End of Linear 
Radiator. 

95. This condition is approximately realized by the shadows 
of the electrodes of a luminous arc with vertical electrodes. 

Let, in Fig. 83, 2 r1 = diameter of the 
lower electrode, I = length of the linear- 
radiator, and 2 w the diameter of the radiator. 
Neglecting first the diameter 2 w of the radi- 
ator, the part of the radiator which, in the 
direction <£, is shaded, is 

s = r, cot <£, (18) 

and the reduction factor of the light, or the 
ratio, by which the intensity of light flux of 
the radiator proper (heading I, class (5)), 
/ = 70 sin 0, has to be multiplied, is 


= 1 - COt <£, (20) 


LIGHT FLUX AND DISTRIBUTION. 


211 


and the light intensity in the direction <£ thus is 
/= qIQ sin <J) 

= /0 ( 1 — j- cot (f> J sin ^ 

r / • ri 

= /0 (sin $ — j cos 

For values of 0 less than fa, where. 


tan 


T 

-1 , 


(21) 


(22) 


the light flux is zero, that is, complete shadow would exist if there 
were no diffusion, etc. 

If we now consider the diameter, 2 w, of the radiator, we get 
the same distribution of intensity, except in the angle 


where fa' and fa' is given by 


tan fa' = -*- 


w 


and tan fa" = -1— — 

L 


In this narrow angle the light flux fades from the value which it 
has at fa" — and which is the same as given by equation (21) — 
to 0 at fa', while, when neglecting 2w, the intensity would 
become zero at fa by equations (22) and (21). 
As illustrations are plotted in Fig. 84 and recorded in Table IV, 


FIG. 84. 


212 RADIATION, LIGHT, AND ILLUMINATION. 

the distribution of light flux for— =0.25; 0.5; 1; 2, as curves I, 

^ri 

II, III, IV, corresponding to an arc length equal to J, J, 1 and 
2 times the electrode diameter. 

777. REFLECTION. 

96. As rarely the distribution of intensity and the brilliancy 
of the radiator are such as desired, reflection, diffraction, and 
diffusion are used to a considerable extent to modify the distribu- 
tion curve and the brilliancy of the radiator. 

Reflection may be irregular or regular reflection. In irregular 
reflection, the light impinging on the reflector is thrown back 
irregularly in all directions, while in regular reflection the light 
is reflected under the same angle under which it impinges on the 
reflector. The former is illustrated by a piece of chalk or other 
dull white body, the latter by the mirror. 
A. Irregular Reflection. 

Irregular reflection is used in indirect lighting to secure dif- 
fusion and low intrinsic brilliancy of the light source by throw- 
ing the direct light against the ceiling and illuminating by the 
light reflected from white or light colored ceilings. In some 
luminous arcs, the so-called flame carbon arc lamps, irregular 
reflection is used to direct most of the light downward by 
using a small circular reflector — usually hollow — immediately 
above the arc, the so-called " economizer." In this case the 
smoke produced by the arc largely deposits on the reflector 
and thereby maintains it of dull white color, the deposit of most 
flame carbons being calcium fluoride and oxide and thus white. 
In irregular reflection, the reflector is a secondary radiator; 
that is, if <$! = that part of the flux of light of the main radiator 
which is intercepted by the reflector, and a = albedo of the 

reflector (that is, the ratio of reflected 
light to impinging light, or the "ef- 
ficiency of the reflector"), the radiator 
is a generator of the light flux a$r 

As an example may be discussed 
the intensity distribution of a vertical 
FlG- 85- luminous arc L having a circular 

(irregular) reflector R immediately above the arc, as shown in 
Fig. 85. Let 2 aj = angle subtended by the reflector R from 


LIGHT FLUX AND DISTRIBUTION. 213 

the base of the arc, I the (vertical) length of the arc. The 
radius of the reflector then is rl = I tan co, and the light flux 
intercepted by the reflector is calculated in the manner as dis- 
cussed under heading II, Class (2) ; that is, if 70 is the maximum 
or horizontal intensity of the arc L, the intensity in the direc- 
tion (180 deg. - <£) will be 

7 = 70 sin <£. (1) 

The reflector then intercepts the entire light flux issuing from 
the radiator L between 180 deg. and 180 — co, and the part q of 
the light flux issuing between 180 — co and 90 deg., which is 
given by 

r, cot (180 - </>) .A 

q = - - — - — = tan co cot <£; (2) 

i 

hence, the light flux intensity which is intercepted by the reflec- 
tor is 

A = ?/o sin * 

= 70 tan co cos <£, (3) 

and the light intensity issuing into space from the main radiator 
in this angle, - < <£< (180 - co), is 


% sn 
= 70 (sin $ — tan co cos (/>). (4) 

Therefore the light flux intercepted by the reflector within the 
angle (f> = 0 to ^> = co (or rather, <f> = 180 deg. to <f> = 180 — co) is 


within the angle <£ = &> to </> = - is 


$/' = 2 ^70 / g sin2 (j>d(j> = 2 nI0 tan o> / cos ^ sin 

I + cos 2 


7 sin 2 co 


(5) 


214 RADIATION, LIGHT, AND ILLUMINATION. 

and therefore the total light flux intercepted by the reflector is 

*1 * */ + ^l" = <"> 

and the reflected light flux, or light flux issuing from the reflector 
as secondary radiator, is 

4>2 = a^1 = xIQaco, (6) 

where a = albedo. 

As the reflector is a plane circular radiator, its maximum 
intensity is in the downward direction, and is given under head- 
ing I, class (2), as ^ 

V = - = /o«", (7) 

71 

and herefrom follows the intensity of radiation of the secondary 
radiator in any direction <£, 

r - /0" cos <£ 

= 70 aw cos 0. (8) 

The total intensity of radiation of main radiator and reflector 
or secondary radiator combined, in the lower hemisphere, or 

for 0 < (j> < -j is 

Zi 

/ = /' + /" = /o (gin 0 + aaj cos <£); (9) 

and in the upper hemisphere light flux issues only under the 
angle - < $ < K — tu, and is 

Zi 

I = I' = 70 (sin ^ - tan w cos 0). (10) 


FIG. 86. 

For to = 75 deg. = and a = 0.7, the intensity distribution 

12 

is plotted in Fig. 86 and given in Table V. The distribution 


LIGHT FLUX AND DISTRIBUTION. 


215 


curve is of the type characteristic of most flame carbon arc 
lamps. 
Substituting the numerical values in 


(9) gives 
and in (10) gives 


7 = 70 (sin $ + 0.92 cos 
7 = 70 (sin $ - 3.73 cos 


TABLE V. 


Regular: a = 0.6. 

<!>. 

Irregular reflection. 
a = 0.7. 
<a = 75 deg. 

Regular reflection, 
a = 0.7. 
tal = 60 deg. 
w2 = 85 deg. 

Irregular: a' = 0.1. 
&<! = 60 deg. 
to2 = 85 deg. 

2r. 
— 1=1. 

I 

0 

3.68 

0 

0.18 

10 

4.32 

0.70 

0.17 

20 

4.83 

1.47 

0.16 

27 

0 16 

30 

5.19 

2.00 

0.42 

35 

0 80 

40 

5.39 

2.57 

1.19 

45 

1 54 

50 

5.44 

3.06 

1.89 

55 

2 23 

60 

5.46 

3.46 

2.55 

65 

4.11 

3 27 

70 

5.02 

4.74 

3.97 

75 

4.82 

5.32 

4.62 

80 

4.58 

5.86 

5.26 

85 

4.30 

6.34 

5.86 

87 5 

5 18 

4 93 

90 

4.00 

4.00 

4.00 

92 5 

2 00 

2 00 

95 

2.68 

0 

0 

100 

1.34 

105 

0 

B. Regular Reflection. 

97. With regular reflection by a polished reflector or mirror 
as used, for instance, in some forms of luminous arcs, the reflector 
is represented by a second radiator, which has the same shape as 
the main radiator and is its image with regard to the plane of the 
reflector. If a is the albedo of the radiator and 70 the maximum 


216 


RADIATION, LIGHT, AND ILLUMINATION. 


intensity of the main radiator, the maximum intensity of the 
virtual or secondary radiator is a/0. 

The reflector then cuts out of the light flux of the radiator 
that part intercepted by it, and adds to the light flux that part 
of the (virtual) light flux of the secondary radiator which passes 
through the plane of the reflector. 

As example may be considered the intensity distribution of a 
vertical luminous arc of length Z, supplied with a circular ring- 
shaped mirror reflector concentric with and in the plane of the 
top of the arc. Let a)l be the angle subtended by the inner, 
co2 the angle subtended by the outer edge of the reflector, from 


FIG. 87. 

the base of the arc, as diagrammatically illustrated in Fig. 87; 
then the intensity of the light flux of the main radiator 


for 
is 
and for 

is 

where, by (2), 


7 = 70 sin 


7T - 


hence, 

and is zero for 


q2 = tan w2 cot 


cos 


(ID 
(12) 

(13) 
(14) 


7/ = 70 (sin c/) — tan 

(f> > 7T - 0)r 

All the light flux issuing from the main radiator between the 
upper vertical and the angle wt (0 < <£ < 6^) 

1' = 70 sin <f> (15) 

is wasted by passing through the central hole in the reflector. 


LIGHT FLUX AND DISTRIBUTION. 217 

Of the light flux issuing between angle ^ and - from the upper 

2i 

vertical, the part f <ut < <£ < - j 

7,-fc/oSin* (16) 

is wasted by passing through the hole in the reflector. 
Since, by (13), 

ql = tan ^ cot <£, (17) 

it is : /! = 70 tan wt cos <£, (18) 

7T 

for ^i < 9 < o • 

z 

All the light flux issuing between the upper vertical and the 
angle w2, 

7'=70sin<£, (19) 

is received by the reflector, with the exception of that part which 
passes through the hole in the reflector. 

Of the light flux issuing between angle aj2 and - from the upper 

Z 

vertical, the part 

72= q2i0sm<j> 
= IQ tan a)2 cos (j>, 

for*>2<(£<^ (20) 

2i 

is received by the reflector, with the exception of that part which 
passes through the hole in the reflector. 

The total light flux intensity reflected by the reflector, or the 
useful light flux of the virtual or secondary radiator, thus is, if 
a = albedo of the reflector, 

Within the angle -><£ > aj2 from the upper vertical, 
2 

7"= a (I2 — 7X) = aIQ (tan co2— tan ajj cos <£; (21) 
within the angle a)2 >(f> >a>l from the upper vertical, 

I'" = a (P - 7J = «70 (sin <£ - tan ^ cos <£) ; (22) 

and for ^ > <£ > 0, 

/"" = a (/' - 7') = 0; (23) . 


218 RADIATION, LIGHT, AND ILLUMINATION, 


hence, the light intensity of the illuminant, consisting of vertical 
radiator and ring-shaped mirror reflector, for 

0 < 0 < Wj is 

7=70sin<£; (24) 

for Wj< <j>< ^2 is 

/= /'+ i"' = /o { (1 + a) sin <f> - a tan ^ cos <£) } ; (25 


for 


I = P + I" = 70 { sin <j> + a (tan w2 — tan w^ cos <£ } ; (26) 


for 

and for 


7 -= 7/ = 70 (sin <f> — tan a>2 cos <£), 

7 = 0. 
17 n 


(27) 


For ^= 60° = , ^2 = 85°= — , and albedo a= 0.7, the in- 

O OU 

tensity distribution is plotted in Fig. 88 and recorded in 
Table V. 


FIG. 88. 

Substituting the numerical values in the foregoing, we have: 

(24) 7 = 70sin<£, 

(25) 7 = 70 (1.7 sin $ - 1.21 cos <£), 

(26) 7 = 70 (sin <£ + 6.79 cos 0), 

(27) 7 = 70 (sin 0 - 11.43 cos <£). 

98. As it is difficult to produce and maintain completely 
regular reflection, usually some irregular reflection is superim- 
posed upon the regular reflection. 

For the irregular reflection, the reflector is a horizontal plane 
radiator. 


LIGHT FLUX AND DISTRIBUTION. 219 

The light flux reflected by a plane circular reflector subtending 
angles co1 to co2, by (6), is 

$.,= */„ a' K L"i), (28) 

where a' is the albedo of irregular reflection. 

This light flux gives in the lower hemisphere the maximum 
intensity for </> = 0 as 

7." - 70a' (o,2 - a,,), (29) 

and thus the intensity of the irregularly reflected light in the 
direction < is 


72 = /0" cos 


- wj cos (£, (30) 


and this intensity thus adds to that given by equations (24) to 
(27) in the preceding. 

If some light is obstructed by the shadow of the lower elec- 
trode, then the light intensity of the main radiator, /', in the 

lower hemisphere within the angle fa < (/> < - > is reduced to 

2 


(31) 
and becomes zero for </> <</>,, where tan <£t == y as discussed 

I/ 

under heading II, class (2), equations (21), (22), where r{ is the 
radius of the lower electrode. 

Thus, with a linear radiator of length I, a diameter of the 
lower electrode of 2 rv a ring-shaped mirror reflector subtending, 
from the base of the arc, the angles ^ and w2, and of the albedo 
of regular reflection a and the albedo of irregular reflection a', 
the light intensity distribution within is 0 < (f> < (j>v 

I = //(^2- Wl)cos^; (32) 

within (f>1 < (/> < Wj is 

I = 70 (sin £+fy K - ^) -- ^J cos ^; (33) 

within ^ < (j> < a>2 is 

/ = /0 (1 + a) sin (/> + \a (co2 - ^) - atan^ -j lcos(/)|; (34) 


220 RADIATION, LIGHT, AND ILLUMINATION. 

within toy - 


7T . 

<21S 


Jsin (j>+\ a (tan a)2- tan w^) + a' (to2-u^ - ~ 


cos 


within 
and within 


I = I0 (sin $ — tan a>2 cos <£), 
K — to2 < </> < n is 
/ = 0. 


(35) 


(36) 


FIG. 89. 

The distribution curve of such an illuminant is plotted in 
Fig. 89 and recorded in Table V for the values 

^= 60 deg. = ; co2 = 85 deg. =-r', 


a =0.60; a'= 0.10, 


and 


I 


1. 


Substituting the numerical values in the foregoing equations 
this gives ^ = 27 deg. 

(32) 7 = 0.044 70 cos </>, 

(33) 7 = 70 (sin <j> - 0.456 cos <£), 

(34) 7 = 70 (1.6 sin <£ - 1.495 cos <£), 

(35) 7 = 70 (sin <j> + 5.364 cos <£), 

(36) 7 = 70 (sin <j> - 11.43 cos <j>). 


FIG. 90. 

As comparison is given in Fig. 90 the distribution curve of 
the magnetite arc, which is designed of the type of Fig. 89 
for the purpose of giving more nearly uniform illumination in 
street lighting. 


LIGHT FLUX AND DISTRIBUTION. 221 

IV. DIFFRACTION, DIFFUSION, AND REFRACTION. 

99. Many radiators are of too high a brilliancy to permit 
their use directly in the field of vision when reasonably good 
illumination is desired. A reduction of the brilliancy of the 
illuminant by increasing the size of the virtual radiator thus 
becomes necessary. This is accomplished by surrounding the 
radiator by a diffracting, diffusing, or prismatically refracting 
envelope. 

Diffraction is given by a frosted glass envelope, as a sand 
blasted or etched globe; diffusion by an opal or milk-glass 
globe. The nature of both phenomena is different to a consider- 
able extent, and a frosted globe and an opal globe thus are not 
equivalent in their action on the distribution of the light flux. 
This may be illustrated by Fig. 91. 

Let, in Fig. 91, 1A, R represent the light-giving radiator, 
for simplicity assumed as a point, and G represent a diffracting 
sheet, as a plate of ground glass. A beam of light, C, issuing 
from the radiator R is, in traversing the diffracting sheet G, 
scattered over an angle, that is, issues as a bundle of beams D} 
of approximately equal intensity in the middle and fading at 
the edges '. The direction of the scattered beam of light D, that 
is, its center line, is the same as the direction of the impinging 
beam C, irrespective of the angle made by the diffracting sheet 
with the direction of the beam. 

Different is the effect of diffusion, as by a sheet of opal glass, 
shown as G in Fig. 91, IB. Here the main beam of light G 
passes through, as C', without scattering or change of direction, 
but with very greatly reduced intensity; usually also with a 
change of color to dull red, due to the greater transparency of 
opal glass for long waves. Most of the light, however, is irregu- 
larly reflected in the opal glass, and the point or area at which 
the beam C strikes the sheet G becomes a secondary radiator 
and radiates the light with a distribution curve corresponding 
to the shape of (2, that is, with a maximum intensity at right 
angles to the plane of G, as illustrated in Fig. 91, 1 B. 

A point P thus receives from a radiator R, enclosed by a diffract- 
ing globe G, a pencil of light, as shown in Fig. 91, 2 A, and from 
the point P the radiator appears as a ball of light, shown densely 
shaded in 3 A, surrounded by a narrow zone of half light, 


222 


RADIATION, LIGHT, AND ILLUMINATION. 


shown lightly shaded, and in the interior of a non-luminous or 
faintly luminous envelope. 

If the radiator R is enclosed by a diffusing globe, Fig. 91, B2, 
the point P receives light from all points of the envelope G as 


FIG. 91. 


3-B 


secondary radiator, and a ray of direct light from the radiator R. 
From the point P the entire globe G thus appears luminous, and 
through it shows faintly the radiating point R, as sketched in 
3B. 

An incandescent-lamp filament in an opal globe thus is clearly 


LIGHT FLUX AND DISTRIBUTION. 


223 


but faintly visible, surrounded by a brightly luminous globe, 
while an incandescent filament in a frosted globe appears as a 
ball of light surrounded by a non-luminous or faintly luminous 
globe, but the outline of the filament is not visible. 

100. The distribution of light flux thus essentially depends 
on the shape of the diffusing envelope, but does not much depend 
on the shape of the diffracting envelope; that is, a diffracting 
envelope leaves the distribution curve of the radiator essentially 
unchanged, and merely smooths it out by averaging the light flux 
over a narrow range of angles, while a diffusing envelope entirely 
changes the distribution curve by substituting the diffusing globe 
as secondary radiator, and leaves only for a small part of the light 
- that of the direct beam C' — the intensity distribution of the 
primary radiator unchanged. 

Thus, for a straight vertical cylindrical envelope surrounding 
a radiator giving the distribution curve shown in Fig. 92, curve I, 


FIG. 92. 

the distribution curve is changed by diffraction (frosted en- 
velope), to that shown in Fig. 92, curve II, but changed to that 
shown by Fig. 92, curve III, by diffusion (opal envelope). The 
latter consists of a curve due to the transmitted light and of the 
same shape as I, and a curve due to the diffused light, or light 
coming from the envelope as secondary radiator. The latter is 
the distribution curve of a vertical cylindrical radiator, as dis- 
cussed under heading I, class (5). 
The shape of a diffusing envelope thus is of essential importance 


224 


RADIATION, LIGHT, AND ILLUMINATION. 


for the distribution of the light intensity, while the shape of the 
diffracting envelope is of less importance. 


TABLE VI. 


f. 

v 

Clear globe. 

v 

Frosted globe. 

V 

Opal globe. 

n 

5 

U 

in 

5 

1U 

20 

6 

8 

11 

2*5 

7 

12 

m9 

30 

9 

18 

17 

35 

15 

26 

40 

34 

35 

25 

45 

49.6 

43 

50 

50.6 

47.5 

32 

55 

49.6 

48 

60 

47.5 

46 

34.5 

fi7 

43 

42 

U 1 

70 

37 

37 

35 

75 

29 

32 

80 

20 

26 

34 

85 

15 

21 

90 

13.5 

17 

32 

95 

13 

14 

100 

12.5 

13 

30.5 

110 

12 

12 

29.5 

120 

11 

11 

27 

130 

9 

9 

24 

140 

6 

7 

20 

150 

0 

2 

15 

160 

0 

0 

10 

It is obvious that frosted glass does not perfectly represent 
diffraction, but some diffusion occurs, especially if the frosting 
is due to etching, less if due to sand-blasting. Opal glass also 
does not give perfect diffusion, but, in the secondary radiation 
issuing from it, the direction of the horizontal or impinging beam 
slightly preponderates. 

101. Regular or prismatic refraction also affords a means of 
decreasing the brilliancy by increasing the size of the virtual 
illummant, and at the same time permits the control of the 


LIGHT FLUX AND DISTRIBUTION. 


225 


intensity distribution. It probably is the most efficient way, 
as involving the least percentage of loss of light flux by absorp- 
tion. 

For instance, by surrounding the radiator R by a cylindrical 
lens, as shown diagrammatically in Fig. 93, the rays of light may 


^ 


FIG. 93. 

be directed into the horizontal (or any other desired) direction, 
and the entire lens then appears luminous, as virtual radiator. 

Usually in this case, instead of a complete lens, individual sec- 
tions thereof are used, as prisms, as shown in Fig. 94, and this 


FIG. 94. 

method of light control thus called "prismatic refraction," or, 
where the light does not pass through, but is reflected and turned 
back from the back of the prism, "prismatic reflection." 

Such prismatically reflecting or refracting envelopes and shades 
have found an extensive use.