LECTURE XI. 
LIGHT INTENSITY AND ILLUMINATION. 

A. INTENSITY CURVES FOR UNIFORM 
ILLUMINATION. 

102. The distribution of the light flux in space, and thus the 
illumination, depends on the location of the light sources, and on 
their distribution curves. The character of the required illumi- 
nation depends on the purpose for which it is used: a general 
illumination of low and approximately uniform intensity for street 
lighting; a general illumination of uniform high intensity in 
meeting rooms, etc.; a local illumination of fairly high intensity 
at the reading-table, work bench, etc. ; or combinations thereof, 
as, in domestic lighting, a general illumination of moderate inten- 
sity, combined with a local illumination of high intensity. Even 
the local illumination, however, within the illuminated area 
usually should be as uniform as possible, and the study of the 
requirements for producing uniformity of illumination either 
throughout or over a limited area thus is one of the main prob- 
lems of illuminating engineering. 

The total intensity of illumination, i, at any point in space is 
proportional to the light intensity, 7, of the beam reaching this 
point, and inversely proportional to the square of the distance 
I of the point from the effective center of the light source : 


If the beam of light makes the angle <f> with the vertical 
direction, the illumination, i, is thus in the direction <j>, the 
horizontal illumination, that is, the illumination of a horizontal 
plane (as the surface of a table), is 

, 7cos< , 

ih = i cos 0 = 


226 


LIGHT INTENSITY AND ILLUMINATION. 227 

and the vertical illumination, that is, the illumination of a vertical 
plane (as the sides of a room), is 

7 sin 0 
iv = i sin <£ = — —!- •, (3) 

If, then, in Fig. 95, L is a light source at a distance lv above 
a horizontal plane P, then, for a point A at the horizontal dis- 


FIG. 95. 


tance lh from the lamp, L (that is, the distance lh from the point 
B of the plane P, vertically below the lamp L), we have: 


, 

Lv 

and the distance of the point A from the light is 


cos 


hence, the total illumination at point A is 

. 7 cos2 () . 


the horizontal illumination is 


l 


7 cos 


and the vertical illumination is 

. 7 cos2 <f> sin 


(4) 
(5) 

(6) 
(7) 
(8) 


where 7 is the intensity of the light source in the direc- 
tion <j>. 

Inversely, to produce a uniform total illumination, i0, on the 


228 RADIATION, LIGHT, AND ILLUMINATION. 

horizontal plane P, the intensity of the light source must vary 
with the angle </> according to the equation (6) : 

7" 1 2 

(9) 

or, if we denote by 70 the vertical, or downward, intensity of 
the light source, 

A, - V,2; (10) 

hence, 

7 = ^- (11) 

gives the intensity distribution of the light source required to 
produce uniform total illumination i0 on a horizontal plane be- 
neath the light. 

In the same manner follows from (7) and (8) : 

To produce uniform horizontal illumination ihQ on a plane P 
beneath the light source L, the intensity curve of the light source 
is given by 

cos3 </> 

and, to produce uniform vertical illumination iVQ of objects in 
the plane P beneath the light source L, 

7- /0 (13) 


cos" <p sin 0 

Where the objects in the plane P which are to be illuminated 
may have different shapes — as on a dining-table, work bench, 
etc., uniformity of the total illumination, i, is desirable; where 
all the objects which shall be illuminated are horizontal — as 
the surface of a drafting-board — constancy of the horizontal 
illumination ih is desirable, while where vertical objects are to be 
illuminated — as, for instance, to read labels on bottles — con- 
stancy of the vertical illumination iv is desirable. 

By "horizontal illumination" ih is here understood the illumi- 
nation of a horizontal plane, which is due to the vertical compo- 
nent of the total light flux, while the "vertical illumination " 
iv is the illumination of a vertical plane, due to the horizontal 
component of the light flux. 


LIGHT INTENSITY AND ILLUMINATION. 


229 


In Fig. 96, the intensity curves of the light source required 
to give uniform total illumination i0 (11) in a horizontal plane 
are plotted as curves I, II and III; the intensity distribution 
for uniform horizontal illumination iho (12) is plotted as curve 
IV, and the intensity distribution for uniform vertical illumina- 
tion iVQ (13) in the horizontal plane beneath the light source 
is plotted as curve V. For convenience, curves IV and V are 
shown in the upper half of the diagram. The numerical values 
for lv = 1 are recorded in Table I. With increasing angle $, 
the required intensity increases very rapidly, and, as is obvious, 
becomes infinite for (f> = 90 deg. 

TABLE I.— (Figs. 95 and 96.) 
UNIFORM DISTRIBUTION ILLUMINATION CURVES. 


f. 

COS (p. 

Total. 

1 

Horizontal. 
1 

Vertical. 
1 

degrees. 

cos2 <f> 

cos3 <j> 

sin ^ cos2 <f> 

0 

1 

1 

1 

00 

5 

996 

1.01 

1.015 

11.60 

10 

985 

1.03 

1.045 

5.90 

15 

966 

.07 

1.11 

4.30 

20 

940 

.13 

1.20 

3.30 

25 

906 

.22 

1.35 

2.88 

30 

866 

.33 

1.54 

2.66 

35 

819 

.49 

1.82 

2.59 

40 

766 

.70 

2.22 

2.64 

45 

707 

2.00 

2.83 

2.83 

50 

643 

2.43 

3.73 

3.17 

55 

574 

3.03 

5.27 

3.70 

60 

500 

4.00 

8.00 

4.60 

65 

423 

5.59 

13.20 

6.16 

70 

342 

8.35 

24.4 

8.90 

75 

259 

15.10 

58.3 

15.60 

80 

174 

33.00 

190.0 

32.50 

85 

087 

132.00 

152.0 

133.00 

90 

0 

00 

00 

00 

103. Therefore, in the problem, as it is usually met, of pro- 
ducing uniform intensity i0 over a limited area, subtending angle 
2 aj beneath the light source, the intensity of the light source 


230 RADIATION, LIGHT, AND ILLUMINATION. 


FIG. 96. 


FIG. 97, 


LIGHT INTENSITY AND ILLUMINATION. 231 

should follow (11) for 0 < <f> < a>. Beyond <£ = a>, the intensity 
may rapidly decrease to zero — as would be most economical, if 
no light is required beyond the area subtended by angle 2 co. 
This, for instance, is the case with the concentrated lighting of a 
table, etc. However, the intensity beyond (f> = to may follow a 
different curve, to satisfy some other requirements, for instance, 
to produce uniform illumination in a vertical plane. Thus in 
domestic lighting, for the general uniform illumination of a room 
by a single illuminant, the intensity curve would follow equation 
(11) up to the angle a> — if 2 w is the angle subtended by the floor 
of the room from the light source — and for $ > cu the intensity 
curve would follow the equation, 

/ = -4-r, (14) 

sin2 <j> 

which gives uniform illumination in the vertical plane, that is, of 
the walls of the room. 

In Fig. 98 are shown intensity curves of a light source giving 
uniform illumination in the horizontal plane beneath the lamp, 
from 0 to CD, and the same uniform illumination in the vertical 
plane from $ = a> to <j> = 90 deg., as diagrammatically shown in 
Fig. 97; that is, uniform illumination of the floor of a room 
and (approximately) its walls, by a lamp located in the center 
of the ceiling, where cu is the (average) angle between the 
vertical and the direction from the lamp to the edge of the 
floor: 

I for co = 30 deg.; or diameter of floor -f- height of walls = 

2 
2 tan 30 deg. = —==1.15. 

II for co = 45 deg.; or diameter of floor -j- height of walls = 

2 tan 45 deg. = 2. 

III for aj = 60 deg.; or diameter of floor -H height of walls = 

2 tan 60 deg. = 2 V3 = 3.46. 

IV for a) = 75 deg.; or diameter of floor -f- height of walls = 

2 tan 75 deg. = 7.46. 

These curves are drawn for the same total flux of light in the 
lower hemisphere, namely, 250 mean hemispherical candle power; 


232 RADIATION, LIGHT, AND ILLUMINATION. 

or, 1570 lumens. The vertical or downward intensities 70 are 
in this case: 

I: aj = 30 deg.; 70 = 428 cp. 

II: cu = 45 deg.; 70 = 195 cp. 
Ill: w = 60 deg.; 70 = 95 cp. 
IV: aj = 75 deg.; 70 = 41.5 cp. 

The values are recorded in Table II, in column I, for equal 
downward candle power 70, and in column a, for equal light flux, 
corresponding to 1 mean hemispherical candle power. 

TABLE II. — (Figs. 97 to 99.) INTENSITY CURVES. 

Uniform illumination from vertical <t> = 0 to <f> = « degrees from verti- 
cal, and 

(a) Uniform illumination (on vertical plane) from <f> = w to horizontal 
0 = 90 deg. 

(6) No illumination beyond <J> = w. 

70 for unity illumination at 0 = 0. 

a and 6 for mean hemispherical candle power 1, or 2 TT lumens. 


<j>. 

ca = 30 deg. 

w = 45 deg. 

a> = 60 deg. 

to = 75 deg. 

/0< 

a. 

6. 

V 

a. 

b. 

V 

a. 

b. 

I0- 

a. 

6. 

0 
5 
10 

.00 
.01 
1.03 

1.71 
1.72 
1.76 

3.73 
3.76 
3.83 

1.00 
1.01 
1.03 

0.78 
0.79 
0.80 

1.57 
1.58 
1.62 

1.00 
1.01 
1.03 

0.38 
0.385 
0.39 

0.67 
0.67 
0.685 

1.00 
1.01 
1.03 

0.166 
0.168 
0.172 

0.222 
0.224 
0.229 

15 
20 
25 

.07 
.13 
.17 

1.83 
1.93 
2.00 

3.98 
4.20 
4.35 

1.07 
1.13 
1.22 

0.83 
0.88 
0.95 

1.68 
1.77 
1.99 

1.07 
1.13 
1.22 

0.41 
0.43 
0.465 

0.71 
0.75 
0.81 

1.07 
1.13 
1.22 

0.178 
0.188 
0.203 

0.237 
0.251 
0.271 

30 
35 
40 

1.20 
1.01 
0.81 

2.05 
1.73 
1.38 

4.47 
3.57 
2.24 

1.33 
1.49 
1.70 

1.03 
1.16 
1.32 

2.08 
2.33 
2.66 

1.33 
1.49 
1.70 

0.51 
0.57 
0.65 

0.89 
0.99 
1.13 

1.33 
1.49 
1.70 

0.221 
0.248 
0.283 

0.295 
0.331 
0.377 

45 
50 
55 

0.67 
0.57 
0.50 

1.14 
0.98 
0.85 

0.57 
0 

1.80 
1.70 
1.49 

1.40 
1.32 
1.16 

2.85 
2.50 
1.10 

2.00 
2.43 
3.03 

0.76 
0.93 
1.16 

1.34 
1.62 
2.02 

2.00 
2.43 
3.03 

0.333 
0.405 
0.504 

0.445 
0.540 
0.672 

60 
65 
70 

0.44 
0.41 
0.38 

0.75 
0.70 
0.65 

1.33 
1.22 
1.13 

1.03 
0.95 
0.88 

0.31 
0 

3.60 
3.51 
3.39 

1.37 
1.34 
1.29 

2.40 
2.00 
0.80 

4.00 
5.59 
8.35 

0.665 
0.930 
1.39 

0.887 
1.24 
1.86 

75 
80 
85 

0.36 
0.34 
0.34 

0.61 
0.58 
0.58 


1.07 
1.03 
1.01 

0.83 
0.80 
0.79 

3.21 
3.09 
3.03 

1.22 

1.18 
1.16 

0.20 
0 

12.80 
12.40 
12.10 

2.13 
2.07 
2.02 

2.85 
2.11 
0.67 


90 

0.33 

0.57 

1.00 

0.78 

3.00 

1.14 

12.00 

2.00 

0.11 

LIGHT INTENSITY AND ILLUMINATION. 


233 


These curves in Fig. 98 consist of a middle branch, giving uni- 
form floor illumination, and two side branches, giving uniform 
side illumination, and are rounded off where the branches join. 


FIG. 98. 


Fig. 99 gives the intensity curves for the same angles, w = 30, 
45, 60, and 75 deg., for uniform illumination only in the hori- 


FIG. 99. 


zontal plane beneath the lamp, but no illumination beyond 
this; for <£ > a>, the light flux rapidly decreases. 

The curves in Fig. 99 are also plotted for equal total light flux, 
of 150 mean hemispherical candle power, or 940 lumens. The 


234 RADIATION, LIGHT, AND ILLUMINATION. 

curve, 0, giving (approximately) uniform illumination within an 
angle of 20 deg., or for a> = 10 deg., is added to the set; this curve, 
however, is plotted for one-tenth the light flux of the other 
curves, 94 lumens, or 15 mean hemispherical candle power. 

The vertical or downward intensities 70 are in this case, for 
equal light flux of 940 lumens : 

I: 'a> = 30 deg.; 70 = 500 cp. 
II: a) = 45 deg.; 70 = 235 cp. 
Ill: co = 60 deg.; 70 = 100 cp. 
IV: aj = 75 deg.; 70 = 25 cp. 
0: aj = 10 deg.; 70 = 7000 cp. 

Fig. 99 best illustrates the misleading nature of the polar dia- 
gram of light intensities. It is hard to realize from the appearance 
of Fig. 99 that curves I, II, III and IV represent the same light 
flux, and curve 0 one-tenth the light flux, that is, little more 
than half the light flux of a 16-cp. lamp. 

Curve 0, however, illustrates that enormous light intensities 
can be produced with very little light flux, if the light flux is 
concentrated into a sufficiently narrow beam. This explains 
the enormous light intensities given by search-light beams: for 
oj = 1 deg., or a concentration of the light flux into an angle of 
2 deg. — which is about the angle of divergency of the beam of a 
good search light —we would get 7 = 700,000 cp. in the beam, 
with 15 mean hemispherical, or 7.5 mean spherical, candle power 
light source; and a light source of 9000 mean spherical candle 
power —a 160-ampere 60- volt arc —would thus, when concen- 
trated into a search-light beam of 2 deg., have an intensity in the 
beam of 70 = 210 million candle power, when allowing 75 per cent 
loss of light flux, that is, assuming that only 25 per cent of the 
light flux is concentrated in the beam. 

The numerical values of Fig. 99 are given as b in Table II, for 
equal light flux corresponding to 1 mean spherical candle power. 

B. STREET ILLUMINATION BY ARCS. 

104. To produce uniform illumination in a plane beneath 
the illuminant, a certain intensity distribution curve is required, 
as discussed in A ; for other problems of illumination, correspond- 
ingly different intensity curves would be needed to give the 
desired illumination. 


LIGHT INTENSITY AND ILLUMINATION. 235 

It is not feasible to produce economically any desired distribu- 
tion curve of a given illuminant. Therefore, the problem of 
illuminating engineering is to determine, from the purpose for 
which the illumination is used, the required distribution of illu- 
mination, and herefrom derive the intensity curve of the illumi- 
nant which would give this illumination. Then from the existing 
industrial illuminants, or rather from those which are available 
for the particular purpose, that is selected whose intensity dis- 
tribution curve approaches nearest to the requirements, and 
from the actual intensity curve of this illuminant the illumination 
which it would give is calculated, so as to determine how near it 
fulfils the requirements. 

The intensity curve of the illuminant, required to give the 
desired illumination, depends on the location of the illuminant 
and the number of illuminants used. Thus if, with a chosen 
location and number of light sources, no industrial illuminant 
can be found which approaches the desired intensity curve 
sufficiently to give a fair approach to the desired illumination, 
a different location, or different number of light sources would 
have to be tried. Here, as in all engineering designs which 
involve a large number of independent variables, judgment 
based on experience must guide the selection. If so, practically 
always some industrially available illuminant can be found 
which sufficiently approaches the intensity curve required by 
the desired illumination. 

As example may be discussed the problem of street lighting. 

This problem is : with a minimum expenditure of light flux- 
that is, at minimum cost — to produce over the entire street a 
sufficient illumination. This illumination may be fairly low, 
and must be low, for economic reasons, where many miles of 
streets in sparsely settled districts have to be illuminated. 
This requires as nearly uniform illumination as possible, since 
the minimum illumination must be sufficient to see by, and any 
excess above this represents not only a waste of light flux, but, 
if the excess is great, it reduces the effectiveness of the illumina- 
tion at the places, where the intensity is lower, by the glare of 
the spots of high illumination. 

Uniformity of street illumination thus is of special importance 
where the illumination must for economic reasons be low; while 
in the centers of large cities, or in densely populated districts, 


236 


RADIATION, LIGHT, AND ILLUMINATION. 


as European cities, the relatively small mileage of streets per 
thousand inhabitants economically permits the use of far greater 
light fluxes, and then uniformity, while still desirable, becomes 

less essential. 

TABLE III — (Figs 100 and 101.) 


Intensity: 100 m. sph. cp. 

Illumination: 200 m. sph. cp.; /„ = 20. 

a. 

6. 

c. 

a. 

6. 

c. 

1 

D. C. 

D. C. 

D. C. 

D. C. 

«c 

enclosed 

enclosed 

Magnetite 

Distance. 

enclosed 

enclosed 

Magnetite 

carbon 

carbon 

arc. 

carbon 

carbon 

arc. 

arc. 

arc. 

Clear 

arc. 

arc. . 

Clear 

Clear inner 

Opal inner 

globe. 

Clear inner 

Opal inner 

globe. 

globe. 

globe. 

globe. 

globe. 

4>. 

7. 

7. 

7. 

x = -• 

i. 

i. 

t. 

h 

0 

30 

45 

59 

0 

15 

22.5 

29.5X10~3 

10 

42 

50 

63 

0.2 

22 

24.5 

30.5 

20 

92 

70 

69 

0.4 

46 

33.0 

31.0 

30 

182 

107 

79 

0.6 

73 

42.5 

31.0 

40 

247 

150 

102 

0.8 

73 

44.5 

30.5 

45 

270 

1.0 

67 

40.5 

29.5 

50 

257 

171 

136 

1.2 

53 

35.5 

28.0 

60 

210 

181 

177 

1.4 

39 

30.5 

26.0 

70 

147 

182 

226 

1.6 

30.5 

25.5 

23.5 

75 

122 

181 

243 

1.8 

24.5 

21.5 

21.5 

80 

97 

160 

250 

2.0 

19.0 

18.0 

19.0 

85 

75 

118 

249 

2.5 

11.5 

13.0 

15.0 

90 

65 

89 

197 

3.0 

7.0 

9.5 

12.0 

100 

57 

82 

47 

3.5 

5.0 

7.0 

9.5 

110 

57 

77 

16 

4.0 

3.5 

5.5 

7.5 

120 

60 

68 

5.0 

2.0 

3.2 

4.8 

130 

35 

62 

6.0 

1.2 

2.0 

3.4 

140 

3 

56 

7.0 

1.0 

1.4 

2.5 

150 

17 

8.0 

0.8 

1 .0 

2.0 

9.0 

0.5 

0.8 

1.7 

10.0 

0.4 

0.6 

1 1 

15.0 

0.2 

0.3 

0.5 

20.0 

0. 1 

0. 1 

0.3 

25^0 

0.1 

Q.I 

0^2 

The arc, as the most economical illuminant, is mostly used 
for street lighting. Fig. 100 gives the average intensity curves 


LIGHT INTENSITY AND ILLUMINATION. 


237 


of three typical arcs for equal light flux of 200 mean spherical 
candle power: 


FIG. 100. 


I. The direct-current enclosed carbon arc, with clear inner 
globe: a curve of the character discussed in Fig. 82. II. The 
direct-current enclosed carbon arc, with opal inner globe: a 


0:20 


III 


1,60 


1,00 


0.4 0,2 


08 0,4 06 08 10 12 It 16 18 20 22 24 26 28 


FIG. 101. 


curve of the character discussed in Fig. 92. III. The magnetite 
arc or luminous arc, with clear globe: a curve of the character 
discussed in Fig. 89. The numerical values are recorded in 
Table III, per 100 mean spherical candle power. 


238 


RADIATION, LIGHT, AND ILLUMINATION. 


Herefrom then follows, by equations (6) and (4), the (total) 
intensity, i, in a horizontal plane beneath the lamp, at the 
horizontal distance lh from the lamp, where lv is the height 
of the lamp above this plane (the street). 

These values of illumination, i, are plotted, with x = - as ab- 

LV 

scissas, in Fig. 101 and recorded in Table III for lv = 20, and 
lamps of 200 mean spherical candle power. 

105. With lamps placed at equal distances 4o, and equal 


FIG. 102. 

heights lv, as shown diagrammatically in Fig. 102, the illumina- 
tion of any point A of the street surface is due to the light flux 
of a number of lamps, and not only to the two lamps 1 and 2, 
between which the point A is situated. As, however, the illumi- 
nation rapidly decreases with the distance from the lamp, it is 
sufficient to consider only the four lamps nearest to the point A. 
The illumination of a point A of the street surface, at a horizon- 
tal distance lh from a lamp, 1, then is: 

i = ^ + i2 + ,; + ^ (15) 

where iv iv is, i4 are the illumination due to the lamps 1, 2, 3, 4, 
respectively. 

I j 

Let -r = P and ^==x; (16) 

k lv 

then the directions under which point A receives light are given 
by: 


tan fa = x, 

tan fa = p — x, 

tan fa = p + x, 

tan fa = 2 p — x} 


(17) 


LIGHT INTENSITY AND ILLUMINATION. 


239 


and 


1 lv2 cos2 

/ 


V cos 


cos 


/4 


cos 


(18) 


where Iv I2, 73, I4 are the intensities of the light source in the 
respective directions <f>v <f>2, 


100 120 110 

FIGS. 103, 104. 

Herefrom are calculated the illumination, i, plotted in Figs. 103 
and 104 and recorded in Table IV for lv = 20 ft. ; p = 5, hence 
lho = 100 ft., Fig. 103, and p = 10, hence lho = 200 ft., Fig. 104, 
for equal light flux of 200 mean spherical candle power per lamp. 

As seen, with the same light flux per lamp, the distribution 
curve III of Fig. 100 gives the highest and the curve I the lowest 
intensity at the minimum point midways between the lamps, 
while inversely I gives the highest and III the lowest intensity 
near the lamp; that is, I, the carbon arc with clear inner globe, 
gives the least uniform, and III, the luminous arc, the most uni- 


240 


RADIATION, LIGHT, AND ILLUMINATION. 


form, illumination, while the carbon arc with opal inner globe, 
II, stands intermediate. 

TABLE IV. — (Figs. 101 to 106.) STREET ILLUMINATION. 


tan <h = x 
tan 02 = p — x 


tan <J»3 = 
tan <f>4 = 


X. 

Equal light flux per lamp. 

Equal illumination at minimum. 

p = 10. 

p= 5. 

p = 10. 

p= 5. 

a. 

6. 

c. 

a. 

&. 

c. 

a. 

b. 

c. 

a. 

6. 

c. 

0 
0.2 
0.4 

0.6 
0.8 
1.0 

1.2 
1.4 
1.6 

1.8 
2.0 
2.5 

3.0 
3.5 
4.0 
5.0 

15.8 
22.8 
46.8 

74 
74 
68 

54 
40 
32 

26 
20 
12.7 

8.2 
6.3 
4.9 
4.2 

23.7 
25.7 
34.7 

44 
46 
42 

37 
32 
27 

23 
19.5 
14.6 

11.3 
9 
8 
6.7 

31.7 
32.7 
33.5 

33.5 
33 
32 

30.5 
29 
26.5 

24.5 
22 
18 

15 
13 
11.5 
10.2 

20 
27 
51 

78 
78 
72.5 

59 
45 
37 

32 
29 
25 

29 
31 
40 

50 
52 
48 

43 
38.5 
35 

32.5 
30 

29 

41.5 
42.5 
43 

43 
43 
42.5 

41.5 
40.5 
39 

37.5 
36 
34.5 

38 
54 
111 

176 
176 
162 

128 
95 
76 

62 
48 
30 

20 
15 
12 
10 

35 
38 
52 

66 
69 
63 

55 
48 
40 

34 
29 

22 

17 
13.5 
12 
10 

31 
32 
32.7 

32.7 
32.3 
31.4 

30 

28.4 
26 

24 
22 

18 

15 
13 
11 
10 

8 
11 
20 

31 
31 
29 

24 
18 
15 

13 

11.6 
10 

10 

11 

14 

17 
18 
17 

15 
13 
12 

11 
10.4 
10 

12. 1~2 
12.3 
12.5 

12.5 
12.5 
12.3 

12.1 
11.8 
11.3 

11 
10.5 
10 

Ratio of minimum intensities. 

Ratio of total light fluxes. 

1 

1.60 

2.43 

1 

1.16 

1.38 

5.95 
2.43 

3.75 
1.53 

2.45 
1.00 

4.0( 
1.3* 

) 3.45 2.90 
1 1.19 1.00 

Ratio of maximum to min. ilium. 

17.6 

6.9 

3.3 

3.1 

1.8 1.25 

The ratio of maximum to minimum illumination is : 

p = 10 p = 5 

I. Carbon arc with clear globe: 17.6 3.1 

II. Carbon arc with opal globe: 6.9 1.8 

III. Luminous, or magnetite, arc : 3.3 1.25 


LIGHT INTENSITY AND ILLUMINATION. 


241 


As seen, lower values of p, that is, either shorter distances 
between the lamps, or greater elevation of the lamps above the 
street surface, give a more uniform illumination, so that, for 
p = 5, III gives only 25 per cent intensity variation, while, for 
p = 10, I gives a very unsatisfactory illumination, alternating 
darkness and blinding glare. 


•0=2- 


iO 


100 


140 


160 


r\ 


i 


FIGS. 105, 106. 

In Figs. 105 and 106 are plotted, and recorded in Table III, the 
illuminations for equal minimum intensity midways between the 
lamps, and for equal distances lho = 200 ft., between the lamps, 
for 

p = 5, or lv = 40 ft. height above the street level, Fig. 105. 

p = 10, or 10 = 20 ft. height above the street level, Fig. 106. 

To produce this minimum intensity of 0.1 candle feet, with 
200 feet distance between the lamps, would require the following 
mean spherical candle powers : 


I. Carbon arc with clear globe: 1190 

II. Carbon arc with opal globe: 750 

III. Magnetite arc: 490 


, or 4=40 ft. 
800 
690 
580 


242 RADIATION, LIGHT, AND ILLUMINATION. 

It is interesting to note the great difference in the light flux, 
required to produce the same minimum illumination, for the 
three distribution curves. 

The carbon arc gains in efficiency and in uniformity of illumi- 
nation by increasing the elevation from 20 to 40 ft., while the 
magnetite arc loses in efficiency —due to the greater distance 
from the illuminated surfaces —but makes up for this by the 
gain in uniformity of illumination. 


C. ROOM ILLUMINATION BY INCANDESCENT 

LAMPS. 

106. Let Fig. 107 represent the intensity distribution of an 
incandescent lamp with reflector, suitably designed for approxi- 
mately uniform illumination in a horizontal plane below the 
lamp. Such a distribution curve can, for instance, be produced 


FIG. 107. 

by a spiral filament F (Fig. 108) located eccentric in a spher- 
ical globe G, of which the upper part is clear glass and covered 
by a closely attached mirror reflector R, while the lower part 
is frosted, as shown diagrammatically in Fig. 105. 

With this arrangement, half of the light flux issues directly, 
with approximately uniform intensity in the lower hemisphere, 
from <j> = 0 to <j> = (f>l} and with gradually decreasing intensity 
from $ = <£j to 0 at <£ = <j>2. The other half of the light flux is 


LIGHT INTENSITY AND ILLUMINATION. 


248 


reflected from the mirror, and, due to the eccentric location of the 
filament, the reflected rays are collected into an angle of about 
45 deg. from the vertical, and cross each other, thereby producing 
the intensity maximum 
at 4> = 30 deg. The 
intrinsic brilliancy is 
sufficiently reduced, 
and the distribution 
curve smoothed out, 
by the frosting of the 
globe as far as not cov- 
ered by the reflector. 
The light in the upper 
hemisphere beyond 
</> = (j>2 then is only 
that reflected by the 
frosting. 

The numerical val- 
ues of intensity of Fig. 
107 are recorded in 
Table V. 

The mean spherical 
candle power of the 
lamp is 12.93, or 163 
lumens ; the mean can- 
dle power in the lower 
hemisphere is 20.20, or 
127 lumens, and the mean candle power in the upper hemi- 
sphere is 5.66, or 36 lumens. 

Table V gives the distribution of illumination i in a horizontal 
plane beneath and above the lamp, for different horizontal 
distances lh and the vertical distance lv = 1, by equation (6), 
and the horizontal illumination ih, by equation (7), as discussed 
in A. These are plotted in Fig. 109, for the lower hemisphere 
in the lower, for the upper hemisphere in the upper, curve. 

Assuming now that a room of 24 ft. by 24 ft. and 10 ft. high 
is to be illuminated by four such lamps, located 6 inches below 
the ceiling in such a manner as to give as nearly as possible 
uniform illumination in a plane 2.5 ft. above the floor (the height 
of table, etc.). 


FIG. 108. 


244 RADIATION, LIGHT, AND ILLUMINATION. 


TABLE V. — (Figs. 107 to 109.) 


#. 

7. 

lh 

lv 
tan c£. 

i = 

I 

COS2<£ 

**~ 

/ 

Ik 

X = TV 

i — 
I 
cosV 

fc*- 

/ 

cos3^ 

(Upper 
hemisphere). 

cos3 tf, 

if. 

i'h- 

0 
10 
20 

30 
40 
50 

60 
65 
70 

75 
80 

85 

90 
95 
100 

105 
110 
115 

120 
130 
140 

150 
160 
170 

180 

21.0 
22.2 
24.5 

26.3 
25.5 
22.5 

20.0 
19.0 
18.0 

17.0 
16.0 
15.0 

13.0 
11.0 
9.5 

8.0 
7.0 
5.5 

4.5 
3.0 
2.5 

2.2 
2.0 
2.0 

2.0 

0 

0.176 
0.364 

0.577 
0.839 
1.192 

1.732 
2.144 
2.745 

3.732 
5.671 
11.43 

00 

11.43 
5.671 

3.732 
2.745 
2.144 

1.732 
1.192 
0.839 

0.577 
0.364 
0.176 

0 

21.0 
21.5 
21.7 

19.8 
15.0 
9.3 

5.0 
3.4 
2.15 

1.13 
0.49 
0.11 

0 
0.08 
0.29 

0.53 
0.84 
1.00 

1.12 
1.23 
1.47 

1.66 
1.77 
1.94 

2.0 

21.0 
21.3 
20.4 

17.0 
11.5 
6.0 

2.5 
1.44 
0.74 

0.29 
0.08 
0.01 

0 
0.01 

0.08 

0.14 
0.29 
0.42 

0.56 
0.80 
1.13 

1.43 
1.66 
1.91 

2.0 

o 

0.1 
0.2 

0.3 
0.4 
0.5 

0.6 
0.7 
0.8 

0.9 
1.0 
1.1 

1.2 
1.3 
1.4 

1.5 
1.6 
1.7 

1.8 
1.9 
2.0 

2.5 
3.0 
3.5 

4.0 
4.5 
5.0 

6.0 
7.0 
8.0 

9.0 
10.0 

21.0 

21.25 
21.55 

21.8 
21.6 
20.8 

19.4 
17.7 
15.8 

13.9 
12.2 
10.6 

9.2 
5.1 
7.2 

6.45 
5.8 
5.2 

4.7 
4.2 
3.9 

2.55 
1.8 
1.33 

1.0 
0.8 
0.67 

0.45 
0.33 
0.25 

0.20 
0.17 

21.0 
21.1 
21.3 

21.0 
20.0 
18.5 

16.5 
14.3 
12.3 

10.4 
8.6 
7.2 

6.0 
5.1 

4.2 

3.55 
3.1 
2.65 

2.25 
1.95 
1.7 

1.0 
0.6 
0.37 

0.25 
0.18 
0.13 

0.10 
0.08 
0.05 

2.0 

2.0 

*i'.i 

1.5 

1.35 

1.0 

1.2 

0.68 

Y.08 

0.9 
0.77 
0.63 

0.5 
0.43 
0.38 

0.28 
0.20 
0.17 

0.14 
0.12 

'b'.48 

0.35 
0.27 
0.20 

0.13 
0.10 
0.08 

0.06 

As the illumination in the space between the lamps is due 
to several lamps and thus is higher than that at the same horizon- 
tal distance outside of a lamp, for approximate uniformity 
of illumination, the distance between the lamps must be con- 
siderably greater than twice their distance from the side walls 


LIGHT INTENSITY AND ILLUMINATION. 245 


0*6 


08 


II 


246 


RADIATION, LIGHT, AND ILLUMINATION. 


of the room. Locating thus the lamps, as shown diagrammati- 
cally in Fig. 110, at 5 ft. from the side walls and 14 ft. from each 
other, the (total) illumination in the lines A, B, C, D in the 
test plane 2.5 ft. above the floor is calculated. As this plane is 
7 ft. beneath the lamps, first the illumination curve in a plane 
7 ft. beneath the lamp is derived from that in Fig. 109, by 
dividing the ordinates by 72 = 49, and multiplying the abscissas 
by 7. It is given in Fig. 111. 


FIG. 111. 

The illumination, i, at any point, P, then is derived by adding 
the illumination ia, ib, ic, id of the four lamps a, 6, c, d, taken 
from curve in Fig. Ill for the horizontal distances of point P 
from the lamps : lhg, lhb, lhc, lhd. These component illuminations 
are plotted in Figs. 112 to 115; as A , Ab, Ac, Ad in Fig. 112; as 
Ba, Bb in Fig. 113, etc., and their numerical values, in thousandths 
of candle feet, recorded in Table VI. In Fig. 116 are shown 
the four curves of the resultant direct illumination, superim- 
posed upon each other. 

107. To this direct illumination is to be added the diffused 
illumination G resulting from reflection by ceiling and walls. 

Let: at = 0.75 = albedo of ceiling; 

a2 = 0.4 = albedo of walls; 


(19) 


while the floor may be assumed as giving no appreciable reflec- 
tion: a = 0. The diffused light, then, may be approximated 
as follows: 

The ceiling receives as direct light the light issuing in the upper 
hemisphere, or 36 lumens per lamp, thus a total of 

Lx = 4 X 36 = 144 lumens, (20) 


LIGHT INTENSITY AND ILLUMINATION. 
TABLE VI. — (Figs. 110 to 116.) 


247 


X. 

Aa. 

Ab. 

A. 

Ba 
and 
Bd- 

B. 

ca. 

cb. 

C. 

Da- 

Db 
and 
Dc. 

D. 

0 

353 

71 

746 

180 

690 

244 

43 

600 

245 

43 

600 

1 

406 

74 

813 

200 

738 

276 

44 

645 

317 

49 

691 

2 

438 

76 

852 

218 

782 

306 

44 

681 

395 

55 

785 

3 

442 

78 

866 

234 

822 

332 

45 

714 

441 

62 

845 

4 

438 

80 

875 

244 

852 

348 

45 

737 

440 

70 

866 

5 

429 

80 

880 

247 

870 

353 

45 

746 

429 

79 

880 

6 

438 

80 

903 

244 

882 

348 

45 

754 

440 

90 

919 

7 

442 

78 

922 

234 

880 

332 

45 

750 

441 

101 

949 

8 

438 

76 

936 

218 

866 

306 

44 

737 

395 

114 

939 

9 

406 

74 

927 

200 

850 

276 

44 

723 

317 

125 

896 

10 

353 

71 

898 

180 

838 

244 

43 

710 

245 

135 

859 

11 

298 

67 

875 

162 

830 

209 

42 

696 

184 

143 

835 

12 

247 

63 

870 

144 

825 

180 

41 

690 

144 

144 

825 

13 

flOO 

60 

129 

156 

39 

115 

143 

14 

167 

57 

114 

136 

37 

94 

135 

15 

143 

54 

102 

118 

35 

79 

125 

16 

121 

51 

90 

103 

33 

66 

114 

17 

104 

48 

81 

90 

31 

56 

101 

18 

90 

45 

72 

80 

30 

49 

90 

19 

80 

41 

63 

72 

30 

42 

79 

20 

70 

38 

57 

64 

30 

36 

70 

21 

61 

35 

52 

58 

29 

33 

62 

22 

55 

33 

48 

52 

29 

30 

55 

23 

52 

31 

44 

47 

28 

26 

49 

24 

45 

2P 

40 

43 

28 

23 

43 

and also receives some reflected light from the walls. Thus, 
if ^ = total light flux received by the ceiling, and 3>2 = total 
light flux received by the walls, the light flux received by the 
ceiling is 

^ = L, + b2a23>2, (21) 

where b2 is that fraction of the light flux issuing from the walls, 
which is received by the ceiling. 
And the light reflected from the ceiling thus is : 

$/= afr = a, (L, + b2a2$>2). (22) 

The walls receive as direct light the light issuing from the 
lamps in the lower hemisphere, between the horizontal, <f> = 90 


248 


RADIATION, LIGHT, AND ILLUMINATION. 


deg., and the direction, <£ = a) (Fig. 110), from the lamp to the 
lower edge of the walls. This angle co varies, and averages 
30 deg. for that half of the circumference, PQR (Fig. 110), at 
which the walls are nearest, and 60 deg. for that half, RSTUP, 
for which the walls are farthest, from the lamp. Hence the 


1.0 


0.8 


0.6 


0.4 


0.2 


Act 


Ac 


12 10 


0.8 

B 
0.6 

0.4 

G 
0,2 

BaScd 

BbStc 

6 *c 

B 

B 

G 

Bfc&c 

Bakd 

X 

x- — 

** -^ 

-^ 

-* 

^» 

-•>s 

X 

/ 

\ 

-*- 

^^ 

^^** 

>J 

^ 

_ ~~ 

_ - 

—. •— 

, — • 

- — 

- -, 

~- .. 

•• — 

4 8 12 18 20 2* 

FIGS. 112, 113. 

light flux received by the walls as directed light, from each lamp, 
is 

- I I sin $ d$ + - T / sin $ d$ = 83 lumens; (23) 

Zt/30° ^^60° 

or, a total of L2 = 4 X 83 = 332 lumens. (24) 


LIGHT INTENSITY AND ILLUMINATION. 


249 


In addition hereto, the walls receive some of the light flux 
reflected by the ceiling. The total light received by the walls 
thus is : 

$2 = L2 + bpfr, (25) 

where bl is that fraction of the light flux issuing from the ceiling, 
which is received by the walls. 


1.0 


0.8 


0.6 C 


G 

0.2 Ca 


Cb 

Cd 

0 Cc 


1.0 


Cb 


12 16 20 


0.8 


0.8 D 


O.i 


0.2 


Dbbd 
0 DC 


Da 


FIGS. 114, 115. 
And the light reflected from the walls thus is : 

*,' - a^2 = a2 (L2 + 6^*,). (26) 

It thus remains to calculate the numerical values of bl and by 
Of the light reflected by the ceiling as secondary generator, 
/, a part is obstructed by the floor, a part received by the walls. 


250 


RADIATION, LIGHT, AND ILLUMINATION. 


The floor is a square plane, of the same size, 24 by 24 ft., as the 
radiator, that is, the ceiling, and at the distance 10. The light 
intercepted by the floor can thus approximately be calculated 
as discussed in Lecture X, II, 1, Fig. 82, for circular radiator 
and circular shades, by replacing the quadratic shade and radia- 
tor by circular shades of the same area, r2n = 242, and r = 

13.5, at the same distance I = 10, hence of the ratio : - = 0.74. 

Calculated as discussed in Lecture X, II, 1, the floor receives 
55 per cent and the walls 45 per cent of the light reflected by the 
ceiling. 

Assuming, approximately, that the walls receive the same 
percentage of the light reflected from the ceiling, as the ceiling 
receives of the light reflected from the walls, or 

&2 - &!, (27) 
equations (21) and (25) become: 

^ = L,+ \a2$>2, (28) 

<f>2 = L2+ 6^^; (29) 


hence, 


_L2+ 

2 1 + 


(30) 


and the light reflected from the ceiling is 

$/ = a^ = «A — — * 2 2; 
the light reflected from the walls is 

<L ' dS ^2~^~ biaiLl 


(30) 


hence, substituting the numerical values : 

$/ = 146 lumens and $/ = 144 lumens " 

are the values of light reflected from the ceiling and from the 
walls respectively. 

Of that from the ceiling, the floor receives (1 - 6J = 0.55, 
and of that from the walls, the floor receives ll = 0.45; hence, 


LIGHT INTENSITY AND ILLUMINATION. 


251 


the diffused light on the floor plane, and thus also, sufficiently 
approximate, on the test plane 2.5 ft. above the floor, is 

*.= (! -&,)*,'+&,*,' 

= 145 lumens, 

and as this plane contains At = 576 sq. ft., the flux of diffused 
light per square foot in the test plane, or the diffuse illumination, 

is L = -r = 0.250 foot-candle. 


r.oo 


500 


400 


300 


200 


100 


/ 
£ 

/ 


FIG. 116. 

108. Adding this diffuse illumination, shown as G, to the 
directed illumination, gives the total illumination, i, shown 
as A, B, C, Dm Figs. 112 to 115, and recorded in Table VI. 

From these curves are taken the values of the distance, 
Table VII, at which the total illumination passes 0.600; 0.650; 
0.700, etc., foot-candle, and plotted in Fig. 117. The points of 
equal illumination, then, are connected by curves, and thus 
give what may be called equi-luminous curves, or equi-potential 
curves of illumination. 

These equi-luminous curves are plotted for every 0.05 foot- 
candle, except that the curves 0.875 and 0.925 are added in 


252 RADIATION, LIGHT, AND ILLUMINATION. 

dotted lines. As seen, the illumination is a minimum of 0.600 
in the corners of the room and a maximum of 0.950 at a point 
between the lamps and the center of the room, and is between 
0.800 and 0.950 everywhere except close to the edges of the 
room. 


FIG. 117 


TABLE VII.— (Fig. 117.) EQUI-POTENTIAL CURVES. 


t. 

A. 

B. 

C. 

D. 

600 

o 

o 

625 

55 

28 

650 

1 15 rC440)l 

56 

675 

1 80 L12 OOJ 

82 

700 

.20 

2 57 10 70 

1 09 

725 

70 

3 45 8 85 

1 37 

750 
775 
800 
825 

.05 
.37 

.77 
1.21 f (620)1 

1.24 
1.88 
2.45 r(576)l 
3.08 Ll2 OOJ 

5.50 7.00 
6.30 

(505) 

1.61 
1.88 
2.18 r(576)l 
2 50 L12.00J 

850 

1.90 Ll2.00J 

3.88 9.00 

3 10 10.35 

875 

4.00 11.00 

5.30 7.47 

4 70 9.50 

900 

5.90 9.90 

6.40 

5.53 8.90 

925 

7.15 9.10 

(634) 

6.13 8.32 

950 

8.20 

7 20 

(686^ 

Cjc\f\\ 

LIGHT INTENSITY AND ILLUMINATION. 253 

The direct light, which reaches the test plane, 2.5 ft. above 
the floor, as directed light from the lamps, issues within the angle 
from the vertical, <£ = 0, up to from <£ = 40 deg. to 0 = 70 
deg., and is 75 lumens per lamp; or, a total of directed light in 
the test plane of 4 X 75 = 300 lumens. The diffused light in 
the test plane is 576 X 0.25 = 144 lumens, and the total light 
in the test plane thus is 444 lumens; while the total light 
issuing from the four lamps is 4 X 163 = 652 lumens, giving an 

444 

efficiency of illumination of ; - = 0.68; or, 68 per cent: the 

652 

444 

average horizontal illumination in the test plane is ihm = — - = 

o7o 

770; while the average total illumination, from Fig. 117, is 
about im = 870. The difference is due to the varying direction 
in which the directed light traverses the test plane. 

Measurement of the illumination of a room by illuminometer, 
to give correct values, thus must take in consideration the 
different directions in which the light traverses every point; 
by measuring the light flux intercepted by a horizontal sur- 
face, the result represents only the horizontal illumination, 
and not the total illumination at the point measured, and 
therefore frequently does not represent the illuminating value 
of the light. 

D. HORIZONTAL TABLE ILLUMINATION BY INCAN- 
DESCENT LAMPS. 

109. Assuming a table, of 5 ft. by 13 ft., to be illuminated 
so as to give as nearly as possible uniform horizontal illumina- 
tion ih. With a light source of the distribution curve, Fig. 107, 
but of four times the intensity, and using two such lamps, they 
would be located vertically above the table, at a distance from 
each other which would be chosen so that, midways between 
the lamps, the illumination is approximately the same as verti- 
cally beneath the lamps. 

In the same manner as discussed in C, the illumination is 
calculated in characteristic lines, indicated as A, B, C in Fig. 
118, using, however, the curve ih of Fig. 109. 

About the most uniform horizontal illumination then is given 
by locating the lamps 5 ft. above the table, 8 ft. from each 


254 RADIATION, LIGHT, AND ILLUMINATION. 

other and 2.5 ft. from the edge of the table, as shown in Fig. 118. 
The illuminations in the lines A, B, and C, and their components 
are plotted in Figs. 119, 120, 121, and recorded in Table VII. 


A — 


1 

( 

f ' T 

(f 

k---J 

B. ^ 

fr-2.5— » 

, 

^ 4 ^ 

v _ 

2J5 

"^ 

2|5 
i 

6 

FIG. 118. 


TABLE VIII.— (Figs. 118 to 121.) HORIZONTAL ILLUMI- 
NATION OF TABLE. 


X. 

Aa. 

Ab. 

A. 

Ba. 

Bb. 

B. 

Ca'b. 

C. 

-2.5 
-2.0 
— 1.5 

2.96 
3.20 
3.36 

0.24 
0.27 
0.32 

3.20 
3.47 
3.68 

2.96 
3.20 
3.36 

0.38 
0.46 
0.48 

3.34 

3.66 
3.84 

1.55 
1.66 
1.79 

3.10 
3.32 
3.58 

— 1.0 
— 0.5 
0 

3.41 
3.37 
3.36 

0.37 
0.43 
0.50 

3.78 
3.80 
3.86 

3.41 
3.37 
3.36 

0.48 
0.50 
0.50 

3.89 
3.87 
3.86 

1.89 
1.96 
1.97 

3.78 
3.92 
3.94 

+ 0.5 
1.0 
1.5 

3.37 
3.41 
3.36 

0.57 
0.67 
0.81 

3.94 
4.08 
4.17 

3.37 
3.41 
3.36 

0.50 
0.48 
0.48 

3.87 
3.89 
3.84 

1.96 
1.89 
1.79 

3.92 
3.78 
3.58 

2.0 
2.5 
3.0 

3.2 
2.96 
2.63 

0.96 
1.15 
1.37 

4.16 
4.11 
4.00 

3.20 
2.96 

0.46 
0.38 

3.66 
3.34 

1.66 
1.55 

3.32 
3.10 

3.5 

2.28 

1.66 

3 94 

+ 4.0 

1.96 

1.96 

3.92 

LIGHT INTENSITY AND ILLUMINATION. 


255 


From the curves as given in Figs. 119 to 121 may then be 
plotted the equi-luminous curves at the table surface, as done 
in Fig. 117 of the preceding paragraph. In this case, which 
represents concentrated illumination, diffusion is not considered, 
but the light is all directed light. 


A 


141 

FIG. 119. 


4.0 


3.0 


2.0 


1.0 


4,0 4.0 


3.0 3.0 


2.0 2.0 


1.0 


/" 

N, 

/ 

\ 

^ 

^ 

"""-- 

\ 

i 

FIG. 120. 


FIG. 121.