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a is the average difference between pairs of
corresponding rings, taken in the sense Cd-arc minus Cd-tube. k is the temperature correction. a′ is a corrected for
temperature change. Ia and It are the
intensities of the arc and tube exposures on a scale of 1 to 20.
Residuals are given in the sense Computed minus Observed, and are
computed from the values of X and b
given for the several groups in the last column. The values of
a, k, a1, and Res. are given in terms of
0.01 mm2 as a unit. For wave length 6438, 0.01 mm2
equals 0.000080 A. U.
Table 2
The Computational Importance of the Collinearity Condition*
p
x′
3x′
1−3x′
δf′x
= fx′ −
fr′
x =
x′+δx′
3x
1−3x
xLx
ΣxLx
p
p′
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1.00
R .000
.000
1.000
1.00
.0000
.000
1.000
1.00
.0000
.0000
1.0000
G .007
.021
.979
1.00
.0070
.021
.979
1.00
.0043
.0162
.4320
.5680
B .993
2.979
1.979
1.00
.9930
2.979
1.979
1.00
.0119
61.30
60.3
.80
r .006
.018
.982
.982
.000
.0000
.0060
.0180
.9820
.982
.0022
.2727
.7273
.727
.741
g .015
.045
.955
.976
.006
.0020
.0130
.0390
.9610
.982
.0080
.0220
.5909
.4091
.721
.622
b .979
2.937
1.937
.979
.003
.0020
.9810
2.9430
1.9430
.983
.0118
44.59
43.59
.723
.683
.60
r .015
.045
.955
.955
.000
.0000
.0150
.0450
.9550
.955
.0055
.4933
.5067
.507
.521
g .023
.069
.931
.951
.004
.0013
.0217
.0651
.9349
.955
.0133
.0304
.7119
.2881
.507
.467
b .962
2.886
1.886
.953
.002
.0013
.9633
2.8899
1.8899
.955
.0116
31.69
30.69
.508
.493
.40
r .034
.102
.898
.898
.000
.0000
.0340
.1020
.8980
.898
.0126
.6995
.3005
.301
.298
g .040
.120
.880
.899
.001
.0003
.0403
.1209
.8791
.898
.0249
.0486
.8290
.1710
.301
.306
b .926
2:778
1.778
.898
.000
.0003
.9257
2.7771
1.7771
.898
.0111
19.03
18.03
.299
.301
.20
r .074
.222
.778
.778
.000
.0000
.0740
.2220
.7780
.778
.0274
.8544
.1456
.146
.154
g .081
.243
.757
.773
.005
.0016
.0794
.2382
.7618
.778
.0490
.0866
.9170
.0830
.146
.131
b .845
2.535
1.535
.776
.002
.0016
.8466
2.5398
1.5398
.778
.0102
9.780
8.780
.146
.144
.10
r .130
.390
.610
.610
.000
.0000
.1300
.3900
.6100
.610
.0481
.9303
.0697
.070
.053
g .130
.390
.610
.623
.013
.0042
.1342
.4026
.5974
.610
.0828
.1397
.9606
.0394
.069
.092
b .740
2.220
1.220
.616
.006
.0042
.7358
2.2074
1.2074
.610
.0088
5.268
4.268
.071
.073
where
ΣxLx
=
rLr+gLg+bLb,
ΣXLx
=RLr+GLg+BLb.
Lr
= 0.370,
Lg
= 0.617,
Lb
= 0.012.
Columns 1 and 2 were taken from Weaver’s data* and represent (1) purity which is one
minus one-hundredth the per cent white given in Weaver’s table;
and (2) the trilinear coordinates (r, g, b) of points
on the 440 mμ wave length line. The point
corresponding to unit purity is designated as (R, G, B)
according to the present notation. The column is headed,
x′, which may represent either r,
g, or b, as indicated. The
g-coordinate, not given in Weaver’s table,
has been calculated from the relation: g = 1
− r − b; which imposes
the second condition, (see introduction).
Column 5 is the test for wave length constancy (Condition 4), and the
variations within triads show that the condition is not quite met, or (1
− 3r)/(1 − 3R) does
not quite equal (1 − 3g)/(1 −
3G) or (1 − 3b)/(1
− 3B).
Each parameter, therefore, defines a distinct point on the 440
mμ wave length line, the point indicated by
r being different from that defined by
g or by b. It is planned, however,
to alter g and b slightly so that they
correspond to the point indicated by r, and to show
that this slight alteration (in general less than 0.002, see Column 7)
is enough to make the calculated purities check.
Column 8 gives the trilinear coordinates of the r-point
correct to four places as is shown by the constancy of the triads in
column 11.
Column 16 gives the values of purity by all three of the Tuckerman forms,
as the headings indicate. The deviations of these purities from those
(Column 1) indicated in Weaver’s table are quite marked and have
already been commented upon by Priest.**
Column 17 gives the purities from the same formulas but by applying them
“blindly” to Weaver’s uncorrected data (Column
2). These are the same values as are plotted on Priest’s
comparison graph.***
The very material reduction of the variation within the triads of Column
16 as compared to that in Column 17 shows the computational importance
of the fourth equation of condition (condition for collinearity with
spectral point and white point).
Tables (2)
Table 1
Data for the Cadmum Line 6438.4696
Plate No.
Etalon Length
a
k
a′
IA
IT
Residuals
140
4.9mm
− 90
− 12
− 102
10
7
− 43
142
“
−149
+28
− 121
10
5
− 3 X =
+30.5
143
“
− 11
− 1
− 12
10
7
+47 b =
−29.8
144
“
+ 38
− 9
+ 29
10
10
− 2
Mean
− 52
24
109
9.8
+ 14
+ 15
+29
8
5
+21
+ 61
−12
+49
12
5
+ 3 X =
−19.6
110
“
−115
+29
−86
3
5
−48 b =
+ 9.36
−104
+71
−33
3
7
+24
Mean
− 10
26
115
19.9
+23
+ 6
+29
8
12
+25
+18
− 4
+14
13
12
− 2
116
“
−16
+ 3
−13
9
14
−15 X =
+13.5
+24
− 5
+19
12
14
+ 10
117
“
−11
+ 9
− 2
12
15
− 8 b =
+2.43
+11
− 8
+ 3
13
15
− 6
118
“
+ 3
− 1
+ 2
12
15
− 4
Mean
+ 7
10
130
29.9
+ 2
+ 5
+ 7
5
10
+ 6
− 9
− 5
−14
12
10
− 7
131
“
+67
−73
− 6
15
9
+ 6 X =
−4.8
−90
+77
−13
13
9
− 3 b =
−1.24
132
“
+16
−13
+ 3
6
10
+ 3
−20
+13
− 7
6
10
− 7
133
“
+62
−41
+21
8
8
− 2
−13
+41
+28
10
8
+ 7 X =
+22.9
134
“
+24
−14
+10
13
9
− 9 b =
− 1.05
+ 6
+14
+20
15
9
+ 3
Mean of all five plates
+ 5
5
a is the average difference between pairs of
corresponding rings, taken in the sense Cd-arc minus Cd-tube. k is the temperature correction. a′ is a corrected for
temperature change. Ia and It are the
intensities of the arc and tube exposures on a scale of 1 to 20.
Residuals are given in the sense Computed minus Observed, and are
computed from the values of X and b
given for the several groups in the last column. The values of
a, k, a1, and Res. are given in terms of
0.01 mm2 as a unit. For wave length 6438, 0.01 mm2
equals 0.000080 A. U.
Table 2
The Computational Importance of the Collinearity Condition*
p
x′
3x′
1−3x′
δf′x
= fx′ −
fr′
x =
x′+δx′
3x
1−3x
xLx
ΣxLx
p
p′
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1.00
R .000
.000
1.000
1.00
.0000
.000
1.000
1.00
.0000
.0000
1.0000
G .007
.021
.979
1.00
.0070
.021
.979
1.00
.0043
.0162
.4320
.5680
B .993
2.979
1.979
1.00
.9930
2.979
1.979
1.00
.0119
61.30
60.3
.80
r .006
.018
.982
.982
.000
.0000
.0060
.0180
.9820
.982
.0022
.2727
.7273
.727
.741
g .015
.045
.955
.976
.006
.0020
.0130
.0390
.9610
.982
.0080
.0220
.5909
.4091
.721
.622
b .979
2.937
1.937
.979
.003
.0020
.9810
2.9430
1.9430
.983
.0118
44.59
43.59
.723
.683
.60
r .015
.045
.955
.955
.000
.0000
.0150
.0450
.9550
.955
.0055
.4933
.5067
.507
.521
g .023
.069
.931
.951
.004
.0013
.0217
.0651
.9349
.955
.0133
.0304
.7119
.2881
.507
.467
b .962
2.886
1.886
.953
.002
.0013
.9633
2.8899
1.8899
.955
.0116
31.69
30.69
.508
.493
.40
r .034
.102
.898
.898
.000
.0000
.0340
.1020
.8980
.898
.0126
.6995
.3005
.301
.298
g .040
.120
.880
.899
.001
.0003
.0403
.1209
.8791
.898
.0249
.0486
.8290
.1710
.301
.306
b .926
2:778
1.778
.898
.000
.0003
.9257
2.7771
1.7771
.898
.0111
19.03
18.03
.299
.301
.20
r .074
.222
.778
.778
.000
.0000
.0740
.2220
.7780
.778
.0274
.8544
.1456
.146
.154
g .081
.243
.757
.773
.005
.0016
.0794
.2382
.7618
.778
.0490
.0866
.9170
.0830
.146
.131
b .845
2.535
1.535
.776
.002
.0016
.8466
2.5398
1.5398
.778
.0102
9.780
8.780
.146
.144
.10
r .130
.390
.610
.610
.000
.0000
.1300
.3900
.6100
.610
.0481
.9303
.0697
.070
.053
g .130
.390
.610
.623
.013
.0042
.1342
.4026
.5974
.610
.0828
.1397
.9606
.0394
.069
.092
b .740
2.220
1.220
.616
.006
.0042
.7358
2.2074
1.2074
.610
.0088
5.268
4.268
.071
.073
where
ΣxLx
=
rLr+gLg+bLb,
ΣXLx
=RLr+GLg+BLb.
Lr
= 0.370,
Lg
= 0.617,
Lb
= 0.012.
Columns 1 and 2 were taken from Weaver’s data* and represent (1) purity which is one
minus one-hundredth the per cent white given in Weaver’s table;
and (2) the trilinear coordinates (r, g, b) of points
on the 440 mμ wave length line. The point
corresponding to unit purity is designated as (R, G, B)
according to the present notation. The column is headed,
x′, which may represent either r,
g, or b, as indicated. The
g-coordinate, not given in Weaver’s table,
has been calculated from the relation: g = 1
− r − b; which imposes
the second condition, (see introduction).
Column 5 is the test for wave length constancy (Condition 4), and the
variations within triads show that the condition is not quite met, or (1
− 3r)/(1 − 3R) does
not quite equal (1 − 3g)/(1 −
3G) or (1 − 3b)/(1
− 3B).
Each parameter, therefore, defines a distinct point on the 440
mμ wave length line, the point indicated by
r being different from that defined by
g or by b. It is planned, however,
to alter g and b slightly so that they
correspond to the point indicated by r, and to show
that this slight alteration (in general less than 0.002, see Column 7)
is enough to make the calculated purities check.
Column 8 gives the trilinear coordinates of the r-point
correct to four places as is shown by the constancy of the triads in
column 11.
Column 16 gives the values of purity by all three of the Tuckerman forms,
as the headings indicate. The deviations of these purities from those
(Column 1) indicated in Weaver’s table are quite marked and have
already been commented upon by Priest.**
Column 17 gives the purities from the same formulas but by applying them
“blindly” to Weaver’s uncorrected data (Column
2). These are the same values as are plotted on Priest’s
comparison graph.***
The very material reduction of the variation within the triads of Column
16 as compared to that in Column 17 shows the computational importance
of the fourth equation of condition (condition for collinearity with
spectral point and white point).