WITH A DESCRIPTION OF A NEW METHOD TO
DETERMINE THE TOTAL REFLECTIVITY OF ANY
SURFACE IN A SIMPLE AND ACCURATE WAY

WITH A DESCRIPTION OF A NEW METHOD TO DETER-
MINE THE TOTAL REFLECTIVITY OF ANY SURFACE
IN A SIMPLE AND ACCURATE WAY

TER VERKRIJGING VAN DEN GRAAD VAN
DOCTOR IN DE WIS- EN NATUURKUNDE AAN
DE RIJKSUNIVERSITEIT TE UTRECHT, OP GEZAG
VAN DEN RECTOR MAGNIFICUS Dr. C. W.
STAR BUSMANN, HOOGLEERAAR IN DE FACUL-
TEIT DER RECHTSGELEERDHEID, VOLGENS
BESLUIT VAN DEN SENAAT DER UNIVERSITEIT
TE VERDEDIGEN TEGEN DE BEDENKINGEN VAN
DE FACULTEIT DER WIS- EN NATUURKUNDE
OP MAANDAG 4 JUNI 1934 DES NAMIDDAGS
TE VIER UUR

Gaarne maak ik van deze gelegenheid gebruik om U, Hoog-
geleerden Ornstein, Hooggeachten Promotor, te danken voor het
vele, waarvoor ik U erkentelijk mag zijn. De onbaatzuchtige wijze,
waarop gij Uwe gaven stelt in dienst der samenleving, zal mij
steeds tot voorbeeld strekken. Indien ik met dit proefschrift een
bijdrage heb mogen leveren tot een van de belangrijke onder-
zoekingen, welke in Uw laboratorium worden verricht, schenkt mij
dat een bijzondere voldoening van mijn werk.

De jaren, dat ik in Utrecht student ben geweest, zullen mij steeds
een aangename herinnering blijven.

Dat het mij daarenboven gegeven is geweest aan boord van
Hr. Ms. ..Willebrord Snelliusquot; een zoo boeiende en interessante
reis naar en door Nederlandsch Indië mede te maken, acht ik een
zeldzaam voorrecht.

Ook de maanden, die ik in Noorwegen mocht doorbrengen, zal
ik niet hcht vergeten.

Tenslotte betuig ik mijn dank aan vermeulen voor vriendschap-
pelijken raad bij het onderzoek en aan schouten, Beezhold en
van der Veen, die allen eenigen tijd bij het werk behulpzaam
waren.

CHAPTER II. Relative observations in different overlapping
regions of the spectrum.

In researches on the relative intensities of spectral lines a
standardised source of light, for which the spectral distribution of
energy is known, is an indispensible instrument (1) ; as such electric
lamps with filaments or strips of tungsten are now in general use.

To calibrate a standard lamp two independent methods have been
worked out and they were found in mutual agreement to within 2
or 3 % in the region from 0.4 ^ to 0.7 ^ (2). The method which is
most readily carried out in practice consists of a determination of
the temperature of the tungsten by means of an optical pyrometer
and of calculating the energy distribution from Planck's formula
and the emissivities i) of tungsten.

Though many investigations have been made on the emissivities,
their results have not in every respect been satisfactory. In the ultra-
violet the values given by different authors differ by no less than
15 % (3). In the visible region Worthing has carried out deter-
minations of the emissivity but at two wave-lengths, 0.665 fA, and
QA67 fi, only (4). Further coblentz and Emerson (5) have made
observations of the reflectivity 2) at room temperature and Weniger
and Pfund (6) investigated the change of reflectivity with tempera-
ture, from which the emissivities can be computed. These obser-
vations however were made with pieces of tungsten which were
carefully polished and it is open to question whether their results
are also valid for tungsten strips as used in standard lamps, which
often have very rough surfaces.

At the Physical Institute of Utrecht University new researches
on the subject were started some years ago in order to supply a
better basis for the calibration of standard lamps by the above

1)nbsp;With quot;emissivityquot; is meant the ratio of the radiation from the tungsten to
the radiation from a black body at the same temperature. This quantity is also
often called the quot;emission factorquot;.

2)nbsp;The quot;reflectivityquot; is the ratio of the intensity of the reflected to that of the
incident light. In other papers this quantity is frequently termed the quot;reflecting
powerquot; or the quot;reflection factorquot;.

According to Kirchhoff's law the emissivity e equals the
coefficient of absorbtion a; and a again is equal to 1—r where r
indicates the total reflectivity. From determinations of the reflectivity
r the emissivity can be computed and this method we have adopted.

The experimental researches were made exclusively on the
reflectivity of tungsten and the value of the observed reflectivities
will be given by the accuracy of the experiments. It should however
not be concluded that the emissivities derived from the reflectivities
have the same value; for the relation e r=l holds only under
certain special conditions and it is very probable that these conditions
will not be fulfilled in all practical cases.

We will return to these theoretical considerations in chapter 5 ;
in chapter 1 to 4, in which the experiments have been described, we
will deal with the reflectivity only.

All determinations were made on tungsten strips such as are
actually used in the tungsten strip lamps manufactured by Philips
at Eindhoven. The tungsten contains 1 % of thorium oxide and the
strips are about 2.5 mm broad and 25 jj, thick.

When tungsten is heated to incandescent temperatures for the
first time the emission is known to change during the first two or
three hours but then to remain practically constant. This process is
called the quot;agingquot; of the tungsten and is explained by a process of
recristallisation.

The tungsten strips used in the experiments had all been aged for
12 hours at least, at a temperature of 2000 or 2500° K and in an
argon atmosphere of 50 cm mercury.

Generally the reflectivity is dependent on the angle of incidence
and the reflected light is polarised, the reflectivities for both com-
ponents being unequal. In most cases however this effect is neghgible

as long as the angle of incidence does not exceed 20°. In the case
of tungsten this follows more specially from researches in which the
emissivity was determined as a function of the angle with the
normal (12).

In the determination of the total reflectivity difficulties arose from
the facts

2.nbsp;that they had no perfectly polished surfaces either. A part of
the light is diffusely reflected and this part it is, according to my
own experience, easy to underestimate.

In the course of the experiments the difficulties mentioned were
overcome in a satisfactory way; a method was developed by which
the total reflectivity could be determined in one single experiment
and independently of the form and condition of the surface (Chap-
ter 1), By this method the total reflectivity was measured in the
region from 0.45 fj. to 1.05 ^ and for strips outside as well as inside
the bulb. Though in appearance the surfaces of the strips varied
from fairly polished to very rough, their total relectivities were
found to agree with each other; real differences between different
strips could not definitely be demonstrated. The following numbers
may convey a general idea of the results arrived at.

In addition to the determinations of total reflectivity experiments
were made giving relative values in different overlapping regions of
the spectrum. By appropriately adjusting these relative measurements
the curve of total reflectivity could be extended into the ultra-violet
down to 0.23 Beyond 1.05 ^ the curve could likewise be extended
by observations of CoBLENTZ and Emerson (5).

The work was completed by researches on the change of reflec-
tivity with temperature made from 0.25 to 1.0/^ and up to 2400° K.

From the data thus collected the emissivity of tungsten was
computed between 0.23and 1,05 fi and from 1000 to 3000° K. The
final results are given in table 18 on page 64 and some curves have
been drawn in figure 21.

A few comparisons with the results of other investigators have
been made in § 3 of Chapter 5. but a full discussion of the subject
has not been attempted.

In a concluding chapter a discussion is given of the accuracy of
a standard lamp calibrated by means of an optical pyrometer and
the conception of a quot;color temperaturequot; is critisised.

The total reflectivity of a surface is defined as the ratio of the
intensity of the total amount of light reflected to the intensity of the
incident light.

We will put this definition into a more mathematical form.
Suppose (fig. 1) the intensity of the light incident along A on a unit

will be reflected; r^s is the specific reflectivity from A to B. The
total amount of light reflected by the unit of area will be

where the integration has to be extended over a hemisphere and
consequently the total reflectivity for direction A is

The usual way in which r^ is determined by experiment consists
of a practical application of formula 1. viz. of measuring the ratio
of the sum of the intensities of the light reflected in different
directions B to that of the incident light. In the following this
method will be denoted as quot;the old methodquot;.

If the reflecting surface is curved and if the reflection is partly
diffuse and partly specular, r^^ will be an irregular function of the
direction B and it will therefore be almost impossible to arrive at
accurate results by using the old method. In such cases we need a
method in which the integration is made experimentally and not by
means of a numerical calculation. Such a method can be based on
the following theorem:

If a surface of opaque material is illuminated homogeneously from
all sides, the ratio of the intensity reflected within a solid angle
dco^ and in a direction A to the intensity incident within an equal
solid angle is equal to the total reflectivity for direction A.

To prove this theorem we will make use of a general optical law
of reciprocity formulated by Helmholtz (7). This law states that
if from light incident along A a fraction r^^ falls in a direction B,
either by transmission, reflection, refraction, diffuse scattering or by
a combination of these processes, from light incident along B the
same fraction r^^ = r^^ will fall in direction A.

Applying this law to a reflecting surface we at once see that the
specific reflectivity from A to B, r^^ must be equal to the specific
reflectivity from B to A, r^^.

Let the amount Idco^ be incident along B, then a quantity
I.dcog. r^^ . dco^ is reflected in direction A and within a solid angle
^ ptb^and if the surface is illuminated homogeneously, the total
amount reflected towards A will be

The intensity incident within a sohd angle du)^ will be I .da)^^
and dividing by this amount we get

r'^is the ratio with which the above theorem is concerned and which
I will call the reflectivity for homogeneous illumination. But since

We thus arrive at a quot;new methodquot; for the determination of a
total reflectivity viz: by illuminating the surface homogeneously
and measuring the reflected light in one direction only.

This new method has very great advantages compared with the
old method; for the total reflectivity is now found in one single
experiment and the accuracy of the observations is entirely indepen-
dent of the form and the condition of the surface. In using the old
method we have, on the contrary, to carry out a whole series of
experiments in each separate case and when the reflection is very
irregular, we can never expect to arrive at accurate results at all.

The investigations carried out on 12 strips of tungsten with
widely different surfaces give full evidence of the value of the
proposed new method; maximum and minimum of total reflectivity
do not differ by more than 3 or 4 % (See § 6 of this Chapter).

The reasonings given above also apply to a more general problem.
Let a beam of light be incident on a piece of milkglass or on any
other piece of turbid material. The light will then be partly trans-
mitted, partly reflected and partly scattered in all directions. The
total fraction of the incident light that is not absorbed by the
material can in this case again be represented by an integral as in
formula 1

We thus arrive easily at the following generahsation of the
theorem given above.

If a piece o[ any material is illuminated homogeneously the ratio
of the intensity of the light leaving the material in a particular
direction A to the intensity of the incident light is equal to 1—a
where a^ represents the total absorbtion for light incident along A.

Under a homogeneous illumination it is thus possible to measure
the total absorbtion of any material in one single experiment. Special
applications of this proposition have not been made in this paper
but I have given the theorem here, since its application may be
of use to others.

We still have to investigate what errors will arise in the proposed
experiments if the illumination is not perfectly homogeneous. In that
case we do not measure r^ as given by formula 3 but

where d^ represents the errors in the homogeneity of illumination.
The total error will consequently be given by

The value of d^ can be determined by experiment for different
directions 5 (See § 4) and in most cases it will be easy roughly to
estimate r^^ ,or better r^^ .from a prehminary experiment. Since we
are dealing with a correction only we need not know r with great
accuracy.

The discussion of errors due to inhomogeneities of the illumina-
tion for the experiments to be described will be found on page 17.

Finally it must here be said that the method to determine a total
reflectivity by homogeneous illumination has already been proposed

in 1920 by Sharp and Little (8). These authors however did not
give a sufficient proof of the general validity of the principle and
the method has not been used except by themselves.

In order to realise a homogeneous illumination the tungsten strip
was mounted at the centre of a cylinder of tin-plated iron, 45 cm in
diameter and 52 cm high (fig. 2). This cylinder and all objects
inside it were painted white with a zinc-white paint, made after a

receipt of the Bureau of Standards (9). The interior was illuminated
by two 100 watt electric lamps arranged as shown in figure 2.
one above and one underneath the tungsten strip. The arrangement
was such that no light could fall directly from the lamps on the
strip ; the latter was illuminated by the light diffusely reflected from
the walls of the cylinder only and it was proved by experiment
(§4) that this illumination was sufficiently homogeneous.

Through a hole in the cylinder the strip was focussed on the slit
of a double monochromator by a lens L. The radiation was measured
by a photocel and amplifier placed behind the monochromator. In
this way ;be could measure the intensity of the light reflected from
an area of the strip limited by the slit of the monochromator, within
a solid angle fixed by the lens L and of a given spectral range.

To determine the total reflectivity we have to compare the intensity
of the reflected light with the intensity of a beam of the incident light
limited in exactly the same way. This was done by reading the
deflection of the galvanometer first when the strip was focussed on
the slit of the monochromator and then a second time when the strip
was drawn away into position T (fig. 2). Doing so we compare the
light coming from part A of the cylinder wall and reflected by the
strip with the light coming directly from part B and it was proved
by special experiment that the luminous intensities of parts A and B
were equal (§4). Since moreover the deflection of the galvanometer
was proportional to the intensity (§ 4) the ratio of the two readings
at once gave the total reflectivity of the tungsten strip.

From this short description the principal features of the experi-
ment will, I hope, be clear. In the following section the different
parts of the apparatus will be described in some detail and in § 4 a
discussion is given of the check experiments that were necessary.

In the experiment the strip was to be drawn away into position
T' (fig, 2). To do so the stand carrying the strip was mounted on
hole-slot-plane fittings. The three legs of the stand were each
provided with pieces of brass one with a hole, one with a slot and
one plane ; these fitted on three brass points screwed to the bottom
of the cylinder and so the position of the stand was accurately
fixed. A similar arrangement has often been used throughout the

experiments. To draw the strip away the stand was slightly tilted
(turning on hole and slot) by means of a cord passing through a
small hole in the cylinder.

The stand itself was of a somewhat complicated construction so
that the tungsten strip could easily be adjusted to whatever position
desired. All strips examined were put in the same position and this
was done by the following contrivance. The stand carrying the
shutter S (fig. 2) was mounted on hole-slot-plane fittings and could
be exchanged with another stand carrying a mirror and a small
electric lamp with straight helical filament. The reflected image of
the filament fell in the slit of the monochromator and in this way
hght was thrown backwards from the slit on the tungsten strip;
from this the light was reflected to some part of the cylinder wall
and all strips were mounted so that the centre of the reflected light
fell in the same direction. This direction was fixed by a circle
on a screen outside the cylinder. The angle of incidence was
± 12°.

To arrange the experiments in a convenient way the cylinder
mantle could be entirely removed from its base which consisted of
a wooden disc screwed to the table. On one side the cylinder had
also a large door through which the interior could be inspected and
slight alterations could be made if necessary.

and the monochromator was a quartz double monochromator after
V. CiTTERT (10). Especially in the infra-red this instrument has a
very low dispersion and consequently the spectral range transmitted
was large, up to 0.05Happily however the reflectivity of tungsten
only varies slowly with wave-length so that no serious errors are
to be feared.

The photocel and amplifier were arranged according to a scheme
devised by Dr. CuSTERS at Eindhoven^) ; in this amplifier two
thermionic tubes with high grid insulation are coupled in parallel
in a balanced bridge circuit so that disturbances due to small
variations in the tension of the batteries are ehminated. The grid
potential of one of the tubes is influenced by the photoelectric
current through the photocel in the usual way and the difference
between the plate currents of the tubes is measured with a galvano-
meter. To avoid disturbances caused by the humidity of the atmos-
phere the photocel and the amplifying tubes are placed in a vacuum
chamber. Fuller details cannot be given here but they may be found
in Dr. CusTERs' publication (11) and in the literature cited there.
The apparatus proved satisfactory though, of course, it cost some
time and trouble before everything functioned satisfactorily.

The experiments were carried out on the second floor and in
order to avoid mechanical disturbances we had to hang the galvano-
meter in a Julius' suspension, a measure which proved very effective.

The galvanometer used was a siemens and Halske moving-coil
galvanometer with soft iron core. The deflections of this instrument
were found to be by no means always proportional to the current •
this proportionality existed only when the axis of the galvanometer
was adjusted to a strictly vertical position, but if the axis slightly
deviated from the vertical the errors soon became great and irregular
as may be seen from table 1.

The figures given apply to a set of experiments in which the
current through the galvanometer could be commutated and the
deflections both to the right and to the left were read off.

Consider for instance the left hand side of table 1 ; in the
corresponding experiment the axis of the galvanometer deviated

1) We are indebted to Dr. CuSTERS for much advice on the use and con-
struction of the amplifier.

from the vertical towards the left and under these circumstances the
deflections to the right and to the left are unequal by no less than
15 % when the deflection is about 30 mm, but when the deflection
is increased to 60 mm the difference decreases to 2 %. If the position
of the galvanometer is altered so that the axis deviates towards the
right from the vertical, the strange inequalities of the deflections of
30 mm also reverses as is shown by the right hand side of table 1 ;
whereas in case the galvanometer was properly adjusted the errors
did not exceed 3 pro mille for deflections up to 120 mm. In the
experiments the distance of the reading scale from the galvanometer
was about 120 cm.

How these great deviations can be explained is uncertain; the
results given in table 1 were strictly reproducible and the possibility
that the coil of the galvanometer touched the core is therefore
excluded.

These results may however give warning never to assume the
hnearity of a galvanometer but always to test it by experiment.
During our observations such tests were regularly repeated.

As explained in § 2 the deflection of the galvanometer was read
off first when the strip was focussed on the slit of the monochro-
mator and then a second time when the strip was drawn away into

position T (fig. 2) ; the ratio of these deflections at once gave the
total reflectivity.

In order that this be exactly true three important conditions must
be satisfied viz.

1.nbsp;The deflections of the galvanometer must be proportional to
the intensities.

2.nbsp;The luminous intensities of parts A and B (fig. 2) of the
cylinder wall must be equal.

Each of these conditions was verified by special experiment and
such verifications can, of course, only be made within certain
observation errors. These errors will influence the reliability of the
observations, so a detailed account of the verification experiments
and their accuracy will now be given.

1. The proportionality between deflection and intensity was
checked in the following way. The reflectivity of a strip was
measured first when the total deflection of the galvanometer was
about 10 cm; then the current through the electric lamps illuminating
the interior of the cylinder was decreased so that the deflection of
the galvanometer reduced to 3 cm or so and a second determination
of the reflectivity was made. If the two determinations agree with
each other, the proportionality between deflection and intensity
is proved.

The great advantage of this method lies in the fact that the check
could be repeated at any time during the observations, without
making any changes in the experimental arrangement. The errors
possible with the galvanometer (page 12) necessitate repeated
testing of the proportionality. In the course of time quite a number
of observations were made, a few of which are given in the
following table.

The deflections were read off to 0.1 mm and the differences in
the reflectivities entered in the last column of the table are not
greater than might be expected from reading
errors. Perhaps there is some tendency for the
reflectivity to increase with decreasing deflections.
For instance a combinations of 9 different obser-
vations yielded the result that with a decrease
of the average deflections from 10 to 3.5 cm,
corresponded an average increase in the absolute
Inbsp;value of the total reflectivity of 0.15 %, fixed with

nality between deflection and intensity will be
better for the smaller deflections. Since all
observations were made with deflections of 5 to
10 cm we may conclude that the observed reflecti-
vities are probably somewhat too low. The absolute
error will be of the order of —0.15 %, to which
corresponds a relative error of —0.3 %, the
reflectivity always being about 50 %.

2. Two small silver mirrors which could be
rotated on vertical spindles 2 cm apart were
placed in the cylinder instead of the tungsten strip.
These mirrors were put in turn in two positions as shown in figure 4
and the luminous intensities of A and B were thus directly compared.

Sometimes a difference of 1 or 2 % was observed but such errors
were easily corrected by slightly altering the position of the two
light sources in the cylinder. To illustrate the results of these
experiments, the data for one special case are given in table 3.

Comparison between the luminous intensities of parts A and B
(fig. 2) of the cyhnder walls.

The comparison between A and B was made 5 times altogether
during the experiments. Calculating the average differences, as
done in table 3, the following values were obtained

Total average 0.1 0.25

The differences between the values resulting from different
verifications may be due to the fact that the position of the light
sources in the cylinder was not always sufficiently carefully adjusted.
Upon the whole however the difference between A and B is small

and in the average reflectivity calculated from all observations no
serious systematic error is to be expected. An error of about 0.3 %
either positive or negative is not impossible.

3. To verify the homogeneity of the illumination a silver mirror
of high reflecting power (97% at Q.7fx) was mounted in the
cylinder. For such a mirror the reflectivity is practically independent
of the angle of incidence and the polarisation of the reflected light
is very small (13). Starting from the normal position this mirror
was turned through angles of 10, 20, 30, and 40° both horizontally
to the left and to the right and vertically upward and downward.
In this way the intensities incident at 20, 40, 60, and 80° were
measured in 4 different quadrants. I found the following averages

Up to 40° the illumination was practically homogeneous, but at
greater angles the intensity was about 4 % too low.

It is easily calculated that the error caused by this inhomogeneity
will be —1.6 % for a perfect diffusing surface, following the cosine
law. Experiments described later show that the diffuse reflection of
a tungsten strip is only about 10 % of the total reflectivity. Conse-
quently the error due to the inhomogeneity of illumination is not
greater than —0.16 % and is probably still smaller, since the diffuse
reflection of a tungsten strip does not follow the cosine law.

The error caused by the defect of radiation from the hole in the
cylinder was by a similar reasoning estimated to —0.5 % for a
perfect diffuser and less than —0.05 % for a tungsten strip.

In addition to these observations the total reflectivity of the white
paint was determined. In the region from 0.45^ to 1.0its value
varied irregularly between 90.6 and 93.1 %, but beyond 0.45 fx the
reflectivity rapidly decreases ; at 0.4 fx the value is 70 %. Thus the
check on homogeneity carried out at 0.7 fx will be valid from 0.45 fx

to 1.0 fA,. Occasionally the reflectivity of the tungsten was also
observed at 0.4 ^ but these values are not perfectly reliable.

Finally diffuse scattering or double reflections of the light in the
lens L may cause a systematic error in the observations. To verify
this point the total reflectivity of a quot;black bodyquot; was determined.
If the body is perfectly black and if there is no diffuse scattering of
light in the lens L, the total reflectivity must be zero. I observed a
value of 0.5 % instead and this will be the maximum amount
of light scattered by L. Suppose the total reflectivity of a tungsten
strip is 50 %, the deflections for the direct and the reflected light
being 100 and 50 mm respectively, then by scattering in the lens L
a constant amount not greater than 0.5 mm will be added to both
deflections. If so, we observe a reflectivity of 50.5 : 100.5 = 50.25 %;
the maximum error is 0.25 % in absolute or 0.5 % in relative value.

The black body used in these experiments consisted of a small
cylinder 2 cm in diameter and 2 cm long with a hole in it and
painted dead black on the inner side. If this object was not perfectly
black, the error will have been smaller than the value computed
above.

A summary of the systematic errors treated in this section and
of their origin is given in the following list.

Altogether some of the errors are positive and some negative
and they will partly cancel each other; the systematic error in the
reflectivities will certainly not be greater than 0.5 % in relative or
0.25 % in absolute value.

above are all of the same order of magnitude. If we wish to increase
the accuracy of the observations we should have to refine the
experiment in every detail.

By the method described in the foregoing sections we may even
determine the total reflectivity of a tungsten strip inside a glass or
quartz bulb. We can determine the correction, necessary for the
presence of the bulb, experimentally as will now be shown.

Suppose the bulb to reflect a fraction r, to transmit a fraction t
and to absorb a fraction a of the incident light. To the fraction t
entering the bulb an amount r .t is added by a first reflection, r2. t
by a second reflection etc. (See fig. 5A). The intensity incident on
the strip will thus be

a fraction t of this reflected light will be transmitted by the bulb, to
which again a quantity r is added by a reflection on the outer side
(fig. 5A). The total intensity falling in direction M will thus
amount to

and this quantity is measured in exactly the same way as the total
reflectivity of strips outside the bulb.

In order to find R we must know r and t. To determine r the bulb
was wrapped in a sheet of black paper and a small hole in the paper
mantle was focussed on the slit of the monochromator (See fig. 5D).
For all bulbs examined in this way the reflectivity r was almost
invariably found to be 7 % in the whole region from 0.45 fxto 1.0 /i.

To determine t the bulb was lowered to the position shown in
figure 5C. We then measure the quantity

as is easily understood. Since r is known already, t can be calculated
and since ^ r -f- a = 1, the absorbtion a is also found by the same
method.

Some values for the absorbtion of different bulbs are represented
graphically in figure 6. In a strip lamp the tungsten slightly eva-
porates in the course of time, forming a thin coating on the glass.

The more a lamp has been used, the greater will be the absorbtion
as is clearly demonstrated by the different curves in the figure.
From 0.5 ^ to 0.45 ^ the absorbtion rapidly increases ; in this region
and beyond 0.45 fx a standardised lamp cannot safely be used unless
the absorbtion of the bulb is determined now and again. This is not
the place, however, to discuss this question in detail (See also
page 73).

The accuracy of a determination inside the bulb will, of course, be
less than of a determination outside the bulb. The absorbtion of the
glass may be unevenly distributed, thus causing systematic errors in

the observations. Comparing measurements made inside and outside
the bulb (tables 6A and 6B on page 26) we see, however, that errors
of this kind, if any, have not been of importance.

Another error will arise from the fact that the tungsten strip will
intercept a small fraction of the light which would after reflection
in the glass form part of the incident light (See dotted lines in
figure 5A). Consequently the presence of the strip will, for some
particular directions, cause an error of 7 % in the homogeneity of
the illumination. If this error be in the 10 % diffuse reflection, the
corresponding error in the final result will be small (less than
0.7 %) ; care however must be taken that such be the case. In most
cases conditions could easily be so chosen that no serious errors
were to be feared.

Errors of this type may be avoided by screening the bulb at the
back with a small piece of black paper (fig. SB). The intensity
incident on the strip will then be

the terms r.t, r^.t etc. in the series on page 19 now being absent.
Focussing the strip on the slit of the monochromator we now
measure the quantity

In many cases I have used both methods, with and without a
screen and the results were always found to agree very well with
each other (See table 4).

I also had at my disposal a discarded strip lamp, the bulb of which
was cut along the line C in figure 5A. A tungsten strip was
investigated first with the bulb in position and afterwards when the
bulb was removed. In this way it was verified that the presence of
the bulb did not introduce serious systematic errors in the deter-
minations of total reflectivity (See table 5).

r is the reflectivity, t the transmission factor and a the absorbtion factor
for the bulb.

R', T. and Rquot; are defined by formulae 2, 3, and 5 of this section
respectively.

is the value of the reflectivity calculated from R' and i?2 the value
calculated from Rquot;.

When starting the observations on strips inside the bulb the
author felt serious misgivings as to the results to be expected. As
will be seen in the next section, however, the determinations inside
and outside the bulb agree so well that the trustworthyness of the
above methods can no longer be doubted.

Determinations of total reflectivity were made on 12 different
strips of tungsten which I will number from 1 to 12.

No. 1 and No. 2 were two different parts of the same strip, selected
for having a smooth surface and aged for 24 hours at a temperature
of 2500° K.

No. 3 was a strip from an old strip lamp that had been used for
years in this institute.

No. 4 was also a strip from a discarded lamp. This strip had been
heated to melting temperature, thereby getting a well polished surface.

Strips 1 to 4 had all been exposed to the air for about half a year
before the final measurements were made.

In addition observations were carried out on two strips (No. 5
and No. 6) instantly after the bulb had been removedi). These
strips had been aged for 14 hours at 2000° K; No. 5 had an almost
perfectly polished surface.

No. 7, 8, and 9 were three strips inside a glass, and 10, 11 and
12 three inside a quartz bulb. Most of these lamps had been used
for some time. Strip No, 9 had an extremely rough surface, the
roughest of all the strips investigated.

The observations were made in the region from 0.45 ju to 1.05 fx
for intervals of 0.05 /n. A series of observations was always started
at 0.7 jLi and was concluded by a second determination at the same
wave-length.

Let d denote the difference between these two observations then,
according to the theory of errors, the mean square error of a single
observation is given by the formula

I here wish to express my thanks to Prof. G. HOLST at Eindhoven for
offering the two strip lamps from which these strips were taken.

At 0.05 [X and 1.0 the errors may have been somewhat greater,
the photocel being less sensitive to these wave-lengths.

To reproduce the complete material would be of little value but
in tables 6A, B, and C a few statistics have been compiled to
illustrate the results arrived at. As will be seen from these data, the
observations made on 12 strips with widely different surfaces are in
almost perfect agreement with each other; a better evidence of the
reliability of the method used can hardly be imagined.

mean value of the standard deviation is 0.7% and is therefore
considerably greater than the mean square error of a single obser-
vation. It follows that the differences between the reflectivities of
the various strips cannot be explained by observation errors only,
and the question arises whether the strips differ in absolute value
of reflectivity only or whether they also differ in the relative change
of reflectivity with wave-length. The observations were numerous
enough to setde this question by stastistical methods in the following
way.

For each strip I have calculated the average reflectivity in the
region from OAS fx to 1.0/^; each value entered in table 7 is the
mean of 12 observations and will therefore be practically free from

Indeed we see from this table that the strips differ in absolute
value but the deviations are so small that they can hardly be
considered as being real; the standard deviation is 0.5 % only. In
the determinations inside the bulb systematic errors of this order of
magnitude may certainly be expected and 4 of the strips measured
outside the bulb had been exposed to air for half a year; it is not
impossible that their surfaces were somewhat dusty or slightly
oxidized.

Comparing table 7 with the data, given at the beginning of this
section, it is apparent that no clear relation exists between the
absolute value of the reflectivity and the condition of the surface:

NO. 1 and 2 are two different parts of the same strip; No. 4 and 5
had very smooth surfaces and No. 9 had a specially rough one.

Average value of reflectivity in the region 0.45,. to 1.0,t for different
strips of tungsten.

To see whether the strips also differed in the relative change of
reflectivity with wave-length, the observations for each strip were
multiplied by such factors that the averages of table 7 were reduced
all to the same value. Data applying to the observations, after these
reductions had been effected, are given in table 8.

Data applying to the observations on 12 different strips after reduction to
the same average value.

0.5
0.3^

0.35

0.4
0.4

0.8

We see that the standard deviations are now reduced to about
0.4 %, that is to the mean square error of a single observation. We
may draw the conclusion that differences in the relative curves, if
any, are much smaller than the observation errors and can therefore
not be demonstrated.

As final values for the total reflectivity of tungsten I have adopted
the averages of all observations on 12 strips. According to the last
column in table 6C these averages are fixed with a mean square
error of about 0.2 % in absolute or 0.4 % in relative value. By the
results of § 4 a relative systematic error of 0.5 % may also exist.

The final results of the observations will be found in table 13 on
page 42. In figure 7 the reflectivities of 4 different strips have been
plotted to give a further demonstration of the mutual accordance of
the observations.

Experimental arrangement for photographic determinations.
The figure is not in true proportion ; in reality the measures of the
arrangement were as follows;

was very small 8 to 10 mm.
The distance from W to the slit of the spectrograph was about 4 cm.
Angle of incidence 20°.

The light source LS (fig. 8) is focussed by the lens L on the
tungsten strip T, and the reflected light is projected on a diffusing
screen W (smoked magnesium oxide). The intensity of the light
diffusely reflected by W is measured photographically in the

is removed and the source LS and lens L are swung round together
to the position indicated by dotted lines. In the ultra-violet the light
source was a water-cooled hydrogen tube (14) that could not easily
be moved. In that case the spectrograph, the tungsten strip and the
diffuser were mounted on a board and swung round together instead.
All parts that had to be put in two different positions or were to be
removable were mounted on hole-slot-plane fittings ( See page 11).

In the visible region the light source was an electric lamp and an
ordinary Fuess glass spectrograph was used. The observations in
the ultra-violet were made with a small Fuess quartz spectrograph
of low dispersion but high luminosity.

The principle idea of using the diffusing screen W is that, if a
certain constant amount of light falls on the screen, it will always
give the same intensity in the spectrograph, independently of the
form and position of the light patch on the screen. It will then be
immaterial whether the tungsten strip is plane or curved, if only all
light reflected falls on the screen. Whether this condition is satisfied
will depend on the geometrical dimensions of the arrangement and
on the homogeneity of the screen and of the spectrograph. On
several occasions verification experiments were made and errors, if
any, were found to be a few percent only. We need not discuss this
question in detail since errors of this kind will be the same for all
wave-lengths and they will not affect the relative value of the
determinations aimed at.

Since the reflection of the strips is partly diffuse a small fraction
of the reflected light will always fall beside the screen W. For this
reason the observations now under consideration have only relative
and no absolute value.

Special attention was paid to the accuracy of the photographic
method. To compare the incident with the reflected radiation we
need a method to cut down the intensities in some well defined
ratio. This was done by placing sector diaphragms before the lens L
(fig. 9) ; inhomogeneities of the lens etc. were eliminated by
rotating the diaphragms with a velocity of about one revolution in
6 seconds. The result was that the intensity on the photographic
plate fluctuated by say 5 or 10% of its total value and with a

not affect the density on the
plate. The times of exposure
varied from 30 seconds to
8 minutes.

The rotating sector dia-
phragms were used as
standard method, but besides
this the spectrograph was
fitted with a wedge shaped
slit. The intensity on the
plate then gradually increa-
ses from one side of the
spectrum to the other, and
if we measure the density

Drawing a smooth curve the irregularities due to the grains of
the plate average out, while at the same time the densities are found
for a set of different intensities. In this way the most efficient use
is made of the space available on a photographic plate.

The intensities for different points (A, E, C) of the curve will
be approximately given by the corresponding widths of the wedge
slit. It depends however on the quality of the spectrograph and
many other factors to what degree this approximation is exact. I

have therefore preferred to cahbrate the wedge sht by means of the
rotating sectors. The results of some of these observations are given
in the following table.

The values in each column have been multiplied by such factors that their
sum is 100.0.

For the glass spectrograph the agreement between slit-width and
intensity is almost perfect but the quartz spectrograph shows
deviations that are by no means negligible. For this reason a
calibration of the wedge slit was always carried out before a set of
determinations in the ultra-violet were begun.

On several occasions I have taken 10 to 13 identical exposures
on one plate in order to test the accuracy of the photographic method
in general. The densities were determined with the Moll self
recording microphotometer of the Utrecht Institute. From the data
thus obtained the mean square error of a single density was
computed by the known formula

for Ilford Process Panchromatic plates £ = 1.0 %
These figures are valid for a density of about 50 %.

The above results were reproducible at different times and the
errors in density were observed to be independent of wave-length.

Since the slope of the density-intensity curve varies with wave-
length, to a constant error in the density different errors in the
intensity will correspond, as is shown in the following table.

On an Ilford Process Panchromatic plate and by a single point
of the curve (fig. 10) the corresponding intensity is fixed with the
mean square errors given in this table; for Empress plates the
errors are about half as great. If we have to compare two different
intensities, the errors will be a factor V'2 greater, but if on the other
hand the curve be measured in different points and if the number
of exposures be increased, the accuracy will increase correspondingly.
From these data it will be understood that under suitable circum-
stances, viz. continuous spectra and constant light sources, photo-
graphic plates may yield accurate results.

The errors so far dealt with are the combined effect of incon-
stancies of the light source, inaccuracies of the plate and errors in
the density determination. The last have been separately investigated
by taking two identical records of the same plate at the same wave-
length directly after one another. Denoting the differences between
corresponding densities read off from both records by d)^, the mean

error computed in this way was 0.5 to 0.6 % for the Empress plates.
Comparing this with the value given on page 32, we see that the
error in the density determination and the total error of photo-
graphic method are equal to each other. Probably the most important
error is made in drawing the smooth curve (fig. 10) and if so, these

errors are due to the graininess of the plate. The above result then
demonstrates that the errors of the photographic method are mainly
caused by the graininess of the plates but not by inhomogeneity
of the plate at large.

Whether these conclusions are wholly justified I cannot say, but
all results given above were found reproducible at least in two
independent sets of experiments. Perhaps some of these questions
are worth a further study.

As has already been said Ilford Process Panchromatic plates and
Ilford Empress plates were used, the former in the visible and the
latter in the ultra-violet region. The plates were developed with
Rodinal 1/20 or with a Metol Hydrochinon Borax developer (Wel-
lington Handbook) ; fog on the plates was carefully avoided; the
plates were generally rocked also during fixation and after thorough
rinsing with fresh water, they were finally rinsed for 10 minutes in
distilled water. Whether these measures have actually contributed
to the accuracy of the determinations, I feel unable to say with
certainty. However it was found as a general rule that the greater
the attention paid to the treatment of the plates, the greater the
accuracy arrived at.

In the visible region three exposures were made, two of the
direct light and in between these one of the reflected radiation. The
intensities of the direct light were cut down by a rotating sector
diaphragm to 50 % ; the intensities to be compared in the spectro-
graph then stood almost in the ratio 1 :1 to each other. By using
the density-intensity relation given by the wedge slit the accurate
value was easily determined.

In the ultra-violet the number of exposures was increased to 9, 5
of the direct alternating with 4 of the reflected radiation. The direct
exposures were taken alternately with sectors of 50 and 40 % so
that an independent check of the wedge slit was effected. Some-
times the hydrogen tube was not perfectly constant, but by the
combined use of sectors and wedge slit the reflectivities could still be
computed in these cases, though not always without some difficulty.

The observations in the visible part of the spectrum lost their
importance after the more effective methods of Chapter 1 were

developed. In figure 11 the measurements on one of the strips are
given mainly as a demonstration of the accuracy of the methods
adopted.

Relative photographic observations in the visible region compared with
the total reflectivities of Chapter 1.

Comparing the photographic determinations with the total reflec-
tivities of Chapter 1 we see that the difference is about 8 %. This
then must be the amount of light that is not caught on the diffusing
screen W (fig. 8), being lost by diffuse reflection on the strip.

In the ultra-violet the observations have greater importance. Two
sets of experiments were carried out. In October 1932 5 series of
observations were made on 4 different parts of the same strip of
tungsten and these are represented by the dotted lines in figure 12.
This strip had been exposed to the air for about 16 months.

The second series of observations was made in June 1933 on three
different strips but directly after they had been taken from the bulb
in which they had been mounted. In figure 12 these determinations
are given by drawn lines.

The values for the different strips being only relative, we may
multiply them with certain factors and so try to bring them into

closer agreement. From the curves the reflectivities at 0.35, 0.325,
0.30, 0.275, and 0.25 fi were read off and the average of these 5
values was calculated; then the reflectivities for each strip were
multiplied with such factors that the averages reduced all to the
same value.

These reductions being made it was observed that the first set of
observations showed systematic deviations from each other as well
as from the second set; the latter however are in almost perfect
agreement as will be understood from figure 12. It is quite possible

that the strips used in the first experiments were shghtly oxidized
or were spoiled in some other way by being exposed to the air for
so long a period. Otherwise the observed discrepancies are difficult
to explain. Moreover the deviations are so large that the mean
square error of the average calculated for all observations together
is greater than the same error calculated for the second set only.
This being so I have thought it justifiable to discard the first set
of observations entirely.

Data applying to the second set after reducing them to the same
average are given in table 11. Being photographic determination the
agreement is very striking.

Data applying to the relative observations in the ultra-violet region a
reduction to the same average.

§ 2. Relative determinations in the region from 0.578 [x to 0.313 /j,
with photocel and amplifier.

The observations dealt with above do not reach beyond 0.375/t
and the total reflectivities of Chapter 1 were not extended below
0.45 fi. We still need some determinations in the intermediate part
of the spectrum and overlapping both the former regions.

Such observations were made with an arrangement similar to that
described in § 2 of Chapter 1. The cylinder now used was 30 cm in

diameter and 30 cm high ; the inner side was smoked with
magnesium oxide which has a high reflecting power far into the
ultra-violet (15).

As before, the tungsten strip was mounted at the centre of the
cylinder ; the light source was a quartz mercury arc lamp hung from
the lid of the cylinder above the strip. This lamp proved sufficiently
constant without special precautions. The intensities were measured
with monochromator and photocel in the same way as in the
experiments of Chapter 1.

Dimensions and arrangement of the present experiment were such
that condition 2 and 3 of page 14 no longer held and consequently
these determinations have no absolute but relative value only.

In all 8 series of observations were made on 5 different strips and
at 6 wave-lengths. Again as in the foregoing section the data for
each strip were multiplied by such factors as to reduce all strips to
the same average value. In table 12 data concerning the observations,
after the reduction had been made, are given ; as we see from the
table the mutual agreement is very satisfactory.

The mutual relation of these with the other observations, so far
dealt with, will be discussed in the following Chapter.

Data applying to the relative reflectivities from 0.578 ft to 0.313 fi
after reduction to the same average.

By means of the relative determinations the curve of total reflec-
tivity of Chapter 1 can easily be extended into the ultra-violet part
of the spectrum, as has been done in figure 13. The dotted lines
represent the original relative observations of tables 11 and 12. By

FIG. 13.

Extension of the curve of total reflectivity into the ultra-violet.
^ I Original relative observations from table 11 on page 37 and table 12 on page 38.

* I The relative observations adjusted to each other and to the curve of total reflectivity.
X Total reflectivities of Chapter 1.

multiplying these relative curves each with a certain constant factor
they were brought in accordance with each other and with the curve
of total reflectivity.

As is seen from the figure all observations fit well together; the
separate points do not deviate from the smooth curve by more than
0.5 % and it is improbable that the error made in adjusting the
relative curves exceeds this amount.

The relative measurements made by the photographic method lie
somewhat lower as the adjusted curve. As has already been pointed
out on page 30 this is due to the fact that in the experiment of
figure 8 a small part of the diffusely reflected light does not fall on
the diffusing screen W. In adjusting the curve as above, the
assumption has been made that the percentage loss by diffuse
reflection is independent of wave-length. We have reason to believe
that this is really the case. Dependance of wave-length would only
occur if the diffuse reflection was either partly due to double
reflections or to diffraction, both rather improbable suppositions. If
for instance double reflections occurred to an appreciable extent the
total reflection would have been much more diffuse than it was. We
need not discuss the question in detail; the difference between the
curves in figure 13 is small, about 4 %, and even as large an error
as 5 % in this difference will cause an error in the reflectivity of
0.2 % only.

In the ultra-violet the reflectivity has, so far as I know, only been
investigated by Hulburt (16); his results, however, do not agree
at all with the curve of figure 13. He observed a value of 30 %
between 0.4 ju and 0.35 n, gradually decreasing to 15 % at 0.2 fx. To
all probability the discrepancy must be ascribed to some impurity of
the tungsten HuLBURT investigated. He also made determinations
for a great number of other metals and for these his data often fit
well with the results of other investigators.

Later HuLBURT also made observations of the emissivity (17) by
comparing the radiation from a black body with the radiation from
incandescent tungsten. The total reflectivities calculated from his
emissivities agree fairly well with the curve of figure 13 (See also
figure 23 on page 67).

As is seen from figure 13 the reflectivity rapidly increases from
0.25 fx to 0.23 fx,; an extrapolation beyond 0.23 fx cannot be made

We may now turn our attention to the visible and infra-red part
of the spectrum. In these regions determinations at room temperature
have been made by Coblentz and Emerson (5) and these have
been plotted in figure 14 together with my own observations.

From 0.5 jjL to 0.9 fx the values of CoBLENTZ and Emerson lie
somewhat lower, but never by more than 2 %. These differences
may be due to the fact that they have used pieces of tungsten that
were carefully polished but had not been aged. Moreover they have
denoted their tungsten as quot;purequot;, from which might been understood
that this tungsten did not contain thorium oxide. One piece of
tungsten denoted as quot;impurequot; did not show the bend at 1.3^ and
the observations on this piece were not taken into account in
calculating the averages.

At 0.95, 1.0, and 1.05 ^ my observations are practically the same
as those of Coblentz and Emerson and with their data it is there-
fore possible to extend my own curve without any further adjust-
ment. It is, of course, open to question whether this extension is quite

safe; such differences as were found to exist between 0.5^ and
0.9 fA, may also occur in other parts of the spectrum, but since my
observations do not reach beyond 1.05 fj. nothing can be ascertained
on this point.

In a paper on the change of reflectivity with temperature Weniger
and Pfund (6) have made the remark that the minimum at O.S fj,
and the bend in the curve at 1.3^ should disappear for an quot;agedquot;
piece of tungsten. According to my results this remark does not hold
for the minimum at 0.8 jji.

In table 13 I have entered the final values of the total reflec-
tivity at room temperature read off from the smooth curves in

figures 13 and 14 and completed in the infra-red with the data of
coblentz and Emerson. In using the values given in this table it
must always be born in mind that I made my observations only on
strips made by the Philips' Manufactories at Eindhoven. It is not
impossible that strips of other origin will have slightly different
reflectivities.

According to what has been said on page 28 the relative mean
square error of the values given in table 13 is 0.5 % in the region
from 0.45 to 1.05 ^^; perhaps the errors are somewhat greater in
the ultra-violet part of the curve, but in virtue of the figures given
in the last columns of tables 11 and 12 serious errors cannot exist.

First experiments were made with an arrangement as shown in
figure 15 By the lens L^ the hght source LS is focussed on the
strip r of a tungsten strip lamp; a part of the reflected light is
caught by the lens I3 and concentrated on a small piece of ground
glass G. The total intensity of the hght falling on G is measured by

means of the monochromator and photocel placed a short distance
behmd. By way of a reflecting prism P and the lens L^ the source
IS also focussed directly on G. Sj and are two shutters.
The observations were made in the following way. First T was
kept at room temperature and by opening the shutters 5i and in

turn the ratio jld of reflected to direct light was determined. As a
rule this ratio was made about 1 : 1 by placing a diaphragm before
the lens L2.

Next the tungsten strip was heated by an electric current to a
constant temperature. A constant amount of light emitted by T
then falls on G and causes a constant deflection of the galvano-
meter. This constant deflection was simply considered as a change
of zero and from the new zero point the ratio between reflected and
direct light was again determined. The change of this ratio with
temperature will then be equal to the change of reflectivity.

In case the radiation from the strip T should become very strong,
the zero was kept on the reading scale by sending a compensating
current through the galvanometer circuit.

It may happen, of course, that the shift of zero will slightly
influence the sensitivity of the photocel arrangement, but any such
changes are accounted for, since we observe the ratio of the
deflections caused by two different intensities one of which (the
direct light) is constant, the other varying with the reflectivity of
the strip.

The great advantage of the method is that the radiation from T
will not influence the accuracy of the observations, if the temperature
of the strip can be kept sufficiently constant. It was possible to
carry out measurements up to a temperature of 2500° K without
any difficulties.

Nevertheless the experiments with this arrangement did not
answer to our expectations ; observations made with different strip
lamps gave results differing by 10 or 20 %, amounts that could
not possibly be real. These discrepancies were found to be due to
warping of the tungsten strip when heated. The geometric dimen-
sions of the arrangement were such that, in case both ends of the
strip should move over 0.1 mm with respect to each other, this would
cause a displacement of 6 mm of the beam of reflected light over the
lens L3 and we observed indeed that experiments on the same strip
lamp yielded quite different results, when half of the lens L3 was
screened off with either a horizontal or a vertical screen. By pro-
jecting an enlarged image of the strip on a screen and varying the
heating current, movements of about 0.1 mm could also directly be
demonstrated.

Using ordinary strip lamps it will therefore be impossible to
obtain reliable results with the experiment of figure 15 or any
similar arrangement.

During the above experiments I had another opportunity to test
the proportionality between the intensity of the incident light and the
deflection of the galvanometer. On the ground glas G light falls
along two different ways viz. the direct light via L2 and P and the
reflected hght via L^, T, and Lg. The check simply consisted of
observing whether the sum of the deflections caused by the separate
intensities was equal to the deflection caused by the sum of the
intensities ; the shutters and S^ were operated first one after the
other and then both simultaneously.

A check based on this principle can be effected in a very simple
way by the arrangement shown in figure 16.

By one lens L two constant light sources LS and LS' are focussed
on the slit of the monochromator, LS directly and LS' by way of a
reflecting glass plate G ; 5 and S' are two shutters.

The advantage of this method is that we need not make any
changes in the experimental arrangement at all ; the openings of the
lenses etc. remain unaltered and all we have to do is to use the
shutters and vary the electric currents through the light sources LS

and LS\ An accurate check of the proportionahty between a deflec-
tion and an intensity is often a matter of some difficulty; perhaps
the method indicated in figure 16 is one of the best experiments for
the purpose.

The proportionality of deflection and intensity has already been
dealt with on page 14; the experiments now under consideration
confirmed the results given there. Up to a deflection of 10 to 15 cm
the errors did not exceed 2 or 3 pro mille.

In a similar investigation the difficulties caused by warping of
the tungsten strip were also experienced by Weniger and Pfund
(6). In the final experiments they have used a strip lamp of special
construction, but in their paper it is not said how this lamp was made
nor how it was ascertained that the lamp came up to the require-
ments. This is to be regretted, as the results I obtained by a different
method do not agree with their observations and a satisfactory
explanation of the discrepancies cannot be given.

Using the method of Chapter 1 in which the tungsten strip is
illuminated homogeneously, warping of the strip will no longer cause
serious errors; we then measure the total reflectivity independently
of the form and condition of the surface (page 7) and displace-
ments of 0.1 mm or so will not influence the results.

A disadvantage of this method is that the incident radiation is
weak in comparison with the radiation emitted by the tungsten strip
itself at higher temperatures and it is now impossible to eliminate
the radiation from the strip, as was done in the experiment of the
foregoing section. The measurements could therefore not be made
up to high temperatures, except in the ultra-violet region where the

The experiments were made in exactly the same way as described
in § 5 of Chapter 1. To increase the energy of the incident light as
much as possible, the two light sources in figure 2 were replaced by
a 100 watt sodium vapor lamp or a 500 watt electric lamp as used
in projection apparatus. These light sources were placed somewhere
in the cylinder and the illumination of the strip will therefore not
have been quite as homogeneous as in the experiments of Chapter 1.
We have however no reason to expect serious errors, since the

In the ultra-violet the cylinder of § 2 Chapter 2 (smoked with
magnesium oxide) and a mercury arc lamp were used.

Since we are here dealing only with the change of reflectivity with
temperature, the different check experiments of § 4 Chapter 1 are
now superfluous and they were not repeated.

When the tungsten strip is heated the temperature of the bulb
will increase too, and errors in the observations might arise if this
increase of temperature causes changes in the transmission t or the
reflectivity r (See formula 2 on page 19). Special experiments were
made but no such changes could be detected, as was to be expected.
I need not discuss these experiments in detail.

In the observations at high temperatures we have to correct for
the radiation emitted by the tungsten strip itself. To determine this
correction the true source of light (sodium vapor lamp etc.) was
switched off so that the radiation from the strip could be measured.
Here another error may arise. As long as the light source is burning,
the tungsten strip will recieve radiation from all sides and it is not
impossible that the strip is heated a few degrees by this radiation.
If so, the temperature of the strip will slightly decrease as soon as
the source is switched off and for the radiation from the strip too
low a value will be observed; the change of reflectivity will conse-
quently be found too high.

Using the sodium vapor lamp or mercury arc lamp, it was easy to
check whether any errors of the above kind occurred. The radiation
from the strip was measured at a wave-length not emitted by the
source and we observed whether the radiation was influenced by
switching the light source off and on. No such influences could be
detected at all.

Using the 500 watt lamp as hght source the errors were, on the
contrary, quite considerable. This was first deduced from improbably
high values found for the increase of reflectivity with temperature
(45 %) ; later a fall of the temperature of the tungsten strip after
switching off the light source was also distinctly observed.

To avoid these difficulties the observations with the 500 watt
lamp were made at so low a temperature that the radiation from the
strip itself was hardly perceptible. The temperature did not then
exceed 1100° K and to higher temperatures an extrapolation has

been made. How this was done will be discussed in detail when
dealing with the results (page 53).

In the observations made with a sodium vapor lamp or mercury-
arc lamps too, the temperature of the strip was not raised above the
value at which the emission from the strip was about equal in
intensity to the reflected light. With a further rise of temperature
the radiation from the strip rapidly increases and accurate deter-
minations will soon become difficult.

The temperature of the strip was determined by means of a
HolboRN-Kurlbaum pyrometer calibrated in this institute at 0.65^.
To calculate the temperature the emissivities at room temperature,
computed from the total reflectivities of Chapter 1, were used. At
0.65 jjL the reflectivity only increases slowly with temperature and
the error caused by a slightly erroneous value for the emissivity
has not been greater than 10° K. The corresponding error in the
change of reflectivity is entirely negligible.

In the ultra-violet region the emission from the strip itself is very
weak and here the observations could easily be extended up to
2400° K. Measurements were made on 5 different strip lamps each
at different temperatures. To reduce the observations to the same
temperatures a linear interpolation was made; the original data
were plotted in a graph and connected by straight lines, and from
the curves the values were read off at given temperatures. For these
temperatures the averages and the mean square errors of the
averages were calculated; the final results have been entered in
table 14 and a set of curves are reproduced in figure 17.

As we see from the figure, the dispersion of the observations was
large, the separate points spreading from 2 to 6%. The question
arises whether these differences are due to observation errors or
to systematic differences of the strip lamps. At 0.313^ two strip
lamps were examined twice (two drawn and two dotted lines in
figure 17) and these repeated observations seem to indicate syste-
matic differences. It is, however, very likely that these differences
did not really exist. We may better treat this question by a statistical
method.

Percentage increase of reflectivity with temperature. The figure shows 7 series
of observations made on 5 different strip lamps. Wavelength 0.313 fi.

this way we may calculate the standard deviation for repeated
observations on one strip lamp and compare it with the standard
deviation calculated from observations on different strip lamps. If
the latter is considerably greater than the former, it may be deduced
that systematic differences between the strip lamps actually existed.

At 0.254 fi repeated observations were made on 4 strip lamps and
here we have the best chance of arriving at a definite result.
The calculations have yielded the following results.

The standard deviation for observations on one strip lamp is even
somewhat greater than for observations on different lamps and
systematic differences are almost excluded by this result.

As has been stated above the observations at 0.313^ seem to
point in another direction, but they are too few (only 2) to allow
any definite conclusion.

According to the above the spread of the observations, as shown
by figure 17, must be due mainly to observation errors. Presumably
the illumination of the strip was not as homogeneous as I had
expected and warping still caused some errors. The light source too,
a mercury arc lamp, was not as constant as we could have wished.
We have, however, no reason to expect serious systematic errors,
since the final values given in table 14 are averages of observations
on 4 or 5 different strip lamps; the spread of the observations is
wholly accounted for by the mean square errors given in the last
column.

Average of the observations in the ultra-violet showing
a linear increase with temperature.

Plotting the percentage increase of reflectivity against temperature
(fig. 18) we see that our observations within possible errors can be
represented by straight lines through the origin. (The origin has
been taken at room temperature =300° K.)

In the visible and infra-red the same result has been found by
WenigeR and Pfund (6), and worthing too, in a study on the
emissivity of tungsten (4), observed a linear dependence of tempe-
rature at 0.665 jx and 0.467 fx.

From these facts we may draw the conclusion that in the region
from 0.25 n to 4.0 /x the reflectivity of tungsten is a linear function
of temperature.

This is of great importance. Of the straight lines in figure 18 one
point, the origin, is always given and only one other point will
suffice to fix the position of the lines. In the infra-red part of the
spectrum it was not possible to extend the observations above
1100° K, but we are now able to extrapolate to higher temperatures.
From its nature this extrapolation will be somewhat uncertain, the
more so since the value of the above conclusion is diminished by
the fact that my observations do not agree with those of Weniger
and Pfund.

Nevertheless I have in the following assumed the relation between
reflectivity and temperature to be always linear. Without this
assumption I should have had to refrain from computations of the
emissivity for temperatures above 1500° K, except in the ultra-
violet region.

We have now only to study the slope of the straight lines of
figure 18 as a function of wave-length; this slope I have expressed
by the percentage increase of reflectivity corresponding to a rise of
temperature of 1000° K. From each single observation this quantity
can be calculated and we are therefore able directly to compare
measurements made at different temperatures. The averages cal-
culated from all observations are given in tables 15 and 16.

Table 15A contains the observations in the ultra-violet converted
according to our present method. In table \5B the determinations,
made with the 500 watt lamp as light source in the region from
0.4 fx to 0.65 IX and up to about 1400° K, are given. A difficulty expe-
rienced when making these observations was that the white painted
cylinder and all objects inside were heated considerably by the
radiation from the light source. The determination of the temperature
of the tungsten strip was made outside the cylinder and it is doubt-
ful whether the values found are also vahd under the actual con-
ditions of the observations; the increase of temperature caused by

N — Total number of observations.
dR = Increase of reflectivity in O/q.
M. S E. 1. = Mean square error of 1 observation.
M. S. E. 2. = Mean square error of average.

the radiation from the Hght source may not have been the same at
room temperature as at 1400° K. It is improbable however that the
error exceeded 30° K, to which a relative error of 3 % in the observed

change of reflectivity corresponds. This error we may safely neglect,
since the observation errors were many times as large.

In table 15C the result of the observations made with a sodium
vapor lamp is given; these determinations agree very well with the

In the infra-red region the radiation emitted by the strip is very
strong and the observations could not be made for temperatures
higher than 1000 or 1100° K. At these temperatures the strip is
only glowing very faintly and I did not succeed in determining the
temperature by means of a pyrometer with sufficient accuracy. Thus
the observations given in table 16 have a relative value only, but
we may adjust them to the data of table 15 by multiplying with a

certain factor. This factor can be determined at 6 different wave-
lengths (from 0.4^ to 0.65 fj,) ; the average I found to be

We see that in adjusting the values of table 16 a relative mean
square error of 10 % is made. The influence this error has on the
final results will be discussed at the end of this section.

In figure 19 the observations of table 15 and the adjusted values
of table 16 have been plotted together. From 0.4 to 0.65the

determinations do not perfectly agree with each other, but on the
whole the deviations from the smooth curve are not of much
consequence.

As is seen from figure 19 the change of reflectivity with tempe-
rature shows pronounced maxima and minima and comparing with
figure 14 on page 41, we are struck by the fact that maxima and
minima in both figures coincide. Where the reflectivity at room
temperature has a minimum, the increase of reflectivity with tempe-
rature shows a maximum and reversely.

19 we have no observation between 0.254 [x and 0.313 ^^ and it is
quite possible that the curve must be drawn as indicated by the
dotted line. In this region an interpolation will be somewhat uncer-
tain. In the following table I have compared the values read off from
the drawn and from the dotted curve.

Percentage increase of reflectivity per 1000° K. increase of temperature.
Final values read off from the curve in figure 19.

readings; in all probability the error will not exceed 0.5 % except
perhaps at 0.23 [x.

In figure 19 the observations of Weniger and Pfund (6) have
also been plotted and they are seen to disagree with my results
completely. As has already been stated on page 47 the data published
by Weniger and Pfund do not suffice to judge of the reliability
of their determinations so that it cannot be decided to what causes
the discrepancy must be ascribed. It has been mentioned on page 42
that our observations disagree also in quite another respect.

The final results read off from the smooth curve of figure 19
(except below 0.3/i) are given in table 17. With these data and the
reflectivities of table 13 on page 42 the emissivities have been
calculated (See Chapter 5).

In the last column of table 17 the mean square errors have been
entered. In the region from 0.25 /j, to 0.7 jjl I have uniformly adopted
an error of 0.3 % in virtue of the 5^^ and 4quot;^ column in tables 15

separate points with respect to the smooth curve is 0.4 % and we
may suppose the error in the curve itself to be somewhat less.
Perhaps the value of 0.3 % is still too high.

1.nbsp;Observation errors found by multiplying the values in the
column of table 16 by 1.42.

2.nbsp;The relative error of 10 % made in adjusting the relative
observations of table 16 to the absolute data of table 15.

Both are mean square errors and expressed in the same way they
will combine to a total error according to the equation

For instance at 0.8 ^ we have
an observation error of 1.42 X 0.35 = 0.5 %
an error in the adjustment of 0.10 X 7.0 = 0.7 %
and consequently a total error

The errors given in table 17 between 0.7^ and 1.05 ^ have been
computed in this way.

A research, similar to that now carried out for tungsten, has
previously been made with platinum (18).

This metal was then found to recristallize incessantly; whenever
it was heated to incandescent temperatures, the reflectivity at room
temperature changed its value and this process never came to a stop.

In making the experiments with tungsten I had a good oppor-
tunity of checking whether this metal shows any such properties.
The reflectivity at room temperature before and after the strip had
been heated, was determined repeatedly. In the ultra-violet 24
observations of this kind are available and from these I computed
the root mean square difference to 0.7 %. On page 24 the mean
square error of a single observation was deduced to be 0.4 % and
accordingly we might expect a root mean square difference of
0.4 X 1/2 = 0.55 %, if the differences are due to observation errors
only. The above value of 0.7 % is somewhat greater but the diffe-
rence is not of much consequence. Moreover the present observations
were made with a mercury arc lamp, which was not perfectly
constant, and it is quite possible that they were less accurate than

Anyhow we may conclude that recristallisation, if it existed, only
caused minor changes in the reflectivity. In other regions of the
spectrum I have obtained the same result, but there the observations
were less numerous and I will not treat them in detail.

From the data in table 13 and table 17 we can calculate the total
reflectivity as a function of temperature and wave-length. I shall
not give the results of these calculations here, but proceed at once
to the emissivity which is connected to the reflectivity by the equation

This is KirchhOFF's law, which we may only apply when a few
special conditions are fulfilled. KirchhOFF's law is always derived
from a system in thermodynamic equilibrium, in which all absorbed
radiation is transformed into heat and in which loss of heat takes
place from radiation only.

A burning strip lamp, however, is by no means in thermodynamic
equilibrium, and to apply Kirchhoff's law we must introduce two
hypotheses viz.

1.nbsp;that the emission of the tungsten strip is only determined by
the temperature and is not dependent upon its surroundings;

It is not my intention here to discuss whether these conditions
actually are fulfilled, but it is advisable to point out that these con-
ditions are essential if we wish to calculate the emissivity from the
total reflectivity.

surface but from a layer, however thin. It is also possible that a
thin gas layer close to the tungsten surface takes part in the
emission. It is then essential that all material that contributes to the
emission shall be of the same temperature.

impossible to make an estimation of how great the differences in
temperature may be and whether they will cause important errors.

In § 1 of Chapter 1 a method was developed of measuring the
total reflectivity by homogeneous illumination of the surface. This
method we may also regard from a different theoretical point of view.

By a fundamental principle of the theory of radiation a closed
space with black walls in thermodynamic equilibrium is always
filled with homogeneous black body radiation E.^^j,, irrespective of
the nature of bodies which may be contained in the space.

r^j, must be taken in this formula as a separately defined quantity
of which we do not know that it is equal to the total reflectivity. We
only know this if we either apply Helmholtz's law of reciprocity
or KiRCHHOFf's law (See formula 1 on page 60).
Here we have thus three different laws.

Two of these laws are sufficient to prove the third; in deducing
Kirchhoff's law however .we have to make use of Helmholtz's
principle. Hence in my opinion the laws 1 and 3 should be regarded
as the fundamental principles, Kirchhoff's law being a direct result
of these two. This being so, I do not altogether think it right to treat
Kirchhoff's law in the first place, as it is usually done in text books
on the theory of radiation.

In table 18 the emissivities calculated from the reflectivities are
brought together. The calculations were made by means of the data
from tables 13 and 17 and do not need any further elucidation.

In a separate table 19 the mean square errors are given. The
relative error in the total reflectivity at room temperature was 0.5 %
(page 43) and the errors in the increase of reflectivity with tempe-
rature are given in table 17. The last increase with temperature.

Both errors are mean square errors and will therefore give a total
error found from the formula already used, E^ = e^s eg^.

E being the relative error in the reflectivity, we find for the
corresponding error in the emissivity

The error in the ratio of the emissivity for two different wave-
lengths will be found by combining the figures in table 19 according
to the law of squares. In the infra-red region this is, however, not

quite exact. The principal error is here caused by inaccurate adjust-
ment of the relative data of table 16 and this error will have the
same relative value from 0.75 /t to 1.05 jj, for all wave lengths. Con-
sequently the total errors in this region are not entirely independent
of each other and by applying the law of squares we certainly get
too large values for the error in the ratio of two emissivities.

In figure 21 the emissivity has been plotted against wave-length
for some temperatures. As might be expected from figure 19 the
curve gradually changes with increase of temperature.

In figure 22 Worthing's emissivities (4) are compared with the
values that I found.

Worthing made his measurements by means of an optical pyro-
meter with red and blue filter and with a black body made of
tungsten. Up to now his values are in general use and they are
included in the International Critical Tables. As we see the devi-
ations are only small; at 0.467 jx worthlno's values are higher by
a constant amount of 0.01 ; at 0.665 the lines cross each other. It
is not altogether impossible that there may be systematic errors in
Worthing's measurements, but it is equally possible to ascribe the
deviations to a difference in the composition of the tungsten used.
The material I used contained 1 % thorium oxide.

Comparison of the spectral emissivities of table 18 with
observations by Hulbert.

0.50

0.4 5-


	'----- wc - H/^	)RTHING .MAKER
-—^—Slj		, jo. 467 ^ ^jo.6 6 5 ^

Deviations of the same order were also ofund between Coblenz
and Emerson's reflectivities at room temperature and mine (fig. 14).

In figure 23 measurements at 2143° K by HULBURT (17) are
given together with my results. HulbURT's own observations were
only relative, but he adjusted them at 0.467to WORTHINO's
emissivities.

In figure 23 considerable deviations are found, but here too it is
not possible to ascertain if these are due to observation errors or to
differences in the quality of the tungsten. The maximum at 0.4 fx is
also present in HulbuRt's results.

1 will not give any further comparison with other results here. As
in the above cases there will always be two possible explanations of
any differences and we have not much chance of deciding with any
certainty which explanation deserves preference.

§ 1. On the accuracy o[ a standard lamp calibrated by means of
an optical pyrometer.
The cahbration of a standard lamp consists of a determination of
the temperature of the strip by means of an optical pyrometer and
a calculation of the energy distribution from PlANCk's law and the
emissivities.

The mean square errors in the emissivities are given in table 19.
I will now shortly discuss the sources of error mentioned under 2
and 3 ; from a mutual comparison of the possible errors we shall
have to decide whether the emissivities are sufficiently accurate and
by what methods a further increase of the accuracy of a standard
lamp can best be obtained.

The value of c^ is, for the present, of no consequence.
Above 1 have used Wien's law instead of Planck's formula but
as long as 2lt;1.0m and Tlt;3000° K the difference is less than
0.7 %. In the error discussion we may therefore use Wien's law
without any restrictions.

the percentage error in the relative intensity of two wave-lengths
is equal to the difference of the percentage errors in the absolute

Thus if we calculate the latter as a function of 2 and T, the former
can be computed in a very simple way.

We may consider an observation with the pyrometer either as a
determination of the temperature T or as a determination of the
emission E; the errors dT and dE will then be connected by

2.nbsp;by adapting the relative calibration to an absolute standard,
usually given by a black body at the melting temperature of gold.

Errors in the temperature determination by means of a pyrometer
may arise from the following sources.

All errors, except the one mentioned under N«. 5, are such that
they will cause a relative error dEjE in the emission independent of
the temperature. Let us for the present assume that the same also
applies to the error N». 5, it then follows from formula 3 that the
errors dT must increase with the square of the temperature T, so that

In reality this will not be exactly so, but since we are aiming at
a rough estimation of the errors only we need not be concerned with
such details.

Mr. Vermeulen of the Utrecht Institute, who has much experience
in the use of a pyrometer, informs me that the mean square error of
a temperature determination with a holborn-kurlbaum pyrometer
is about 10° K at 2500° K. I will here use this value; if it is either
too large or too small all errors computed below must be reduced
by a corresponding factor.

Inserting this value in equation 3 and expressing I in microns
(10—4 cm) we get

By this simple formula the percentage mean square error in the
relative intensities is expressed independent of temperature.

In table 20 I have compared the errors calculated from equation
6 with the errors caused by inaccuracies in the emissivity. Te latter
were computed as indicated on page 62; they are not independent
of temperature so that the table is valid for 2000° K only.

In most cases the errors Cg are considerably greater than the
errors e^ and the differences increase with the interval between the
two wave-lengths. From this we may draw the conclusion that to
increase the accuracy of a standard lamp, we have in the first place
to effect more accurate determinations of temperature.

If the two wave-lengths lie close together the errors due to the
temperature determination will become very small, but since the
values of table 18 were derived from smooth curves the same will

wave-lengths. (7= 2000 °K).
ei = error due to errors in the emissivities.
ej = error due to errors in the temperature determination.

be the case with the errors in the relative emissivities. It is impossible
to decide which error will be most important in these cases.

On page 69 the absorbtion by the bulb was also mentioned as a
source of serious errors. How the absorbtion can be determined has
been explained in § 5 of Chapter 1 and in figure 6 some absorbtion
curves have been given.

With the magnesium oxide cylinder and mercury arc lamp
measurements of the absorbtion were also made in the ultra-violet

One of the bulbs was not quite clean and shows a strong absorb-
tion. In the region from 0.4 fx to 0.6 fx the observations made with a
mercury arc lamp lie 13^ % lower than the values obtained with the
arrangement of figure 2. This indicates that the absorbtion was not
the same everywhere on the bulb. In the ultra-violet the absorbtion
is seen to increase to no less than 18 %.

The other bulb was quite clear and the absorbtion is almost
constant, about 1 %. from 1.0^ to 0.4/^; here too, however, the
absorbtion increases in the ultra-violet, in this case to 7.0 %.

That serious errors may arise does not need any further demon-
stration ; especially in the ultra-violet they may be many times as
large as the values of table 20.

In virtue of the above we may conclude that to increase the
accuracy of a standard lamp we must

3. Carry out more accurate observations of the increase of
reflectivity with temperature in the region from 0.7 ju to 1.0 fi. In
this part of the spectrum the errors in the emissivities of table 18 are
still quite considerable.

Differentiating formula 1 on page 69 with respect to 2 it is easily
proved that the wave-length of the observation must be fixed within
10 or 20 A. Every worker, however, has this error in his own hand.

Other errors will occur as the theoretical conditions of page 60
are not fulfilled. We have at present no data available from which
to judge the order of magnitude of these errors, so that we can only
surmise that they will be smaller than the other errors dealt with
above.

The quot;color temperaturequot; T^ is defined as that temperature for
which a black body (PlanCK's law) has the same spectral energy
distribution, as has the tungsten at its true temperature T. True
temperature T and color temperature T^ will thus be connected by

The conception of a color temperature has been introduced to
describe the energy distribution of incandescent tungsten by a simple
formula. We see, however, from equation 7 that by the color tempera-
ture the relative emissivities are entirely fixed as a function of wave-
length and a T^ will therefore only be valid within a limited region
of the spectrum.

from which it is seen that, when a color temperature exists, the
logarithm of the emissivity must be a linear function of the inverse
of wave-length.

In figure 25 I have plotted log. e^^ooo against Ijl From this
figure we at once infer that to tungsten a color temperature can be
assigned in the region from 0.4 ^ to 0.7 [x, but outside this region
the conception of a T, does not hold at all. From the slope of the
straight Hne T, was computed to 2023° K (See formula 8).

Conversely we may calculate the emissivities from the color
temperature by reading log. e. j. from the straight line in figure 25.
In table 21 the emissivities thus found have been compared with their

In the region from 0.4 ^ to 0.7^ the agreement is very close, the
differences not exceeding 0.5%, but outside this region the dis-
crepancies vary from 4 to 20 %.

By the graphic method of figure 25 the color temperatures were
computed for different values of the true temperature. The results
of these computations are given in table 22.

Especially for higher temperatures the differences between the
new determinations and the old values (19) are rather large.

Comparison of the emissivities calculated from the color
temperature with their true values from table 18.

ture in the region from 0.75 /a to 1.0 jU. Here we find for T = 2000° K,
7^ = 2150° K, a considerable difference with the value of 2023° K
vahd in the visible region.

It should finally be remarked that from a purely scientific point
of view the conception of a color temperature has no meaning at all.
It has however great practical value, the more so since the color
temperature is valid over the whole visible range of the spectrum.

Photometrie. Sammlung Vieweg.
2 H. C. v. AlPHEN : Dissertation Utrecht. 1927.
3' E.quot; SPILLER: Zs. f. Physik. Bd. 64, 1930, p. 39.

]. W. T. Walsh : Photometry, p. 217.
10 P h V Gittert: Zs. f. Instrumentkunde 41, 1921.
IL J.'h. f. Güsters: Zs. f. technische Physik, 1933, N». 4. p. 154.
12. E. spiller: Zs. f. Physik, 72, 1931, p. 215.

15.nbsp;A. H. Taylor: Joum. of the Optical Society of Amerika. 21, 1933, p. //b.

16.nbsp;O. E. HüLBURT : Astrophysical Journal 42, 1915, p. 205.
17 O E. Hulburt : Astrophysical Journal 45, 1917, p. 149.

W. J. BeekmAN und F. W. OUDT : Zs. f. Physik 33, 1915, p. 831.
19. A. G. Worthing : Astrophysical Journal 61, 1925, p. 146.

Het is gewenscht, dat in leerboeken over en bij het onderwijs in
de waarschijnlijkheidsrekening ook aandacht wordt besteed aan de
methoden en opvattingen der mathematische statistiek.

De foutentheorie van Gausz is gecompliceerd en gaat van te veel
premissen uit; zij wordt daardoor voor den praktischen natuur-
kundige weinig toegankelijk.

F. Möller, Integration der Bewegungsgleichungen im gekrümm-
ten Isobarenfelde.

Het in dit artikel door Möller behandeld probleem behoort tot
de puntmechanica; daarbij te spreken van een isobarenveld is
onjuist.

Dt ~ dxquot;^^ dx 'dy dy ^ dz èz
is onvolledig. De grootheden t]^. rj en rj^ zijn in werkelijkheid drie

componenten van een tensor, waarvan de andere zes componenten
niet zonder meer mogen worden verwaarloosd.

De correspondeerende algemeene vergelijking kan ook langs
zuiver mathematischen weg worden afgeleid.

Th. Hesselberg, Ann. der Hydrographie und mari-
time Meteorologie 1929, H 10, p. 319.

In een één- of meerzijdig begrensden oceaan met stabiel gelaagde
watermassa's kunnen geen stationnaire driftstroomen bestaan, wan-
neer de wind ergens een component heeft loodrecht op de kust-
richting.

Voor de mathematische beschrijving van verschillende problemen
in oceanographie en meteorologie is het gewenscht, dat wordt ge-
zocht naar een algemeen stel vergelijkingen, waarin hydrodynamica
en diffusieverschijnselen tezamen als één probleem worden behandeld.

Het is niet juist, dat de vergelijking van RichaRDSON voor de
thermische electronen-emissie strikt genomen geldt voor een veld-
sterkte nul.

	Axis of the galvanometer deviating from the vertical
Current	to the left		to the right
10-8 Amp.	Deflections		Deflections
	to left	to right	to left	to right
0.4	6.2	6.3	6.2	6,0
0.8	11.8	12.7	12.9	11.8
2.0	26.7	3L0	33.3	28.2
4.0	6L0	62.1	64.0	60.3
5.0	76.8	78.0	78.0	77.1

Wavelength	Deflections in cm		R in %
in II	Direct	Reflected	R in %
0.55	8.33 3.51	4.38 1.85	52.6 62.8
0.75	9.90 6.32 1.59	4.87 3.13 0.78	49.3 49.5 49.1
0.90	9.74 4.43	5.21 2.38	53.5 53.8

Wavelength in /t	A	B	Relative dif- ference in %
0.5	83.9	83.9	0.0
0.6	92.6	91.7	1.0
0.7	94.2	94.1	0.1
0.8	95.8	95.4	0.4
0.9	96.2	95.4	0.8

k	R'	Rquot;	T	r	t	a	Rx	R2
0.5	51.3	50.5	93.7	7.0	89.8	3.2	51.1	53.8
0.6	55.2	51.9	95.1	7.2	90.3	2.5	55.0	54.8
0.7	54.5	52.0	95.8	7.2	90.7	2.1	53.5	54.2
0.8	55.5	52.8	97.0	6.7	91.7	1.6	54.1	54.5
0.9	56.5	54.0	97.5	6.7	92.0	1.3	54.9	55.5

I	Ri	R2	i?3
0.5	49.2	50.9	50.6
0.6	49.7	50.2	(54.4)
0.7	53.6	54.6	53.3
0.8	51.4	52.5	52.5
0.9	52.9	52 6	53.5

Wavelength n	Maximum	Minimum %	Average %	Standard deviation	M. 5. E. of average
6 A. Observations outside the bulb.
0.5	51.0	53.0	51.9	0.6	0.25
0.6	53.4	54.5	53.7	0.4	0.15
0.7	54.3	55.6	54.9	0.5	0.2
0.8	52.8	54.5	53.5	0.6	0.25
0.9	53.4	55.0	54.2	0.6	0.25
1.0	55.8	59.4	57.5	1.1	0.4
6 B. Observations inside the bulb.
0.5	52.0	52.9	52.5	0.3	0.1
0.6	53.5	54.9	54.1	0.6	0.25
0.7	54.2	56.4	55.1	0.9	0.4
0.8	51.8	54.6	53.2	1.1	0.4
0.9	53.4	54.5	53.9	0.4	0.15
1.0	55.5	57.5	56.7	0.8	0.3
6 C. All observations combined.
0.5	51.0	53.0	52.2	0.6	0.15
0.6	53.4	54.9	53.9	0.5	0.15
0.7	54.2	56.4	55.0	0.7	0.2
0.8	51.8	54.6	53.4	0.9	0,25
0.9	53.4	55.0	54.1	0.5	0.15
1.0	55.5	59.4	57.1	1.1	0.3
M.S.E. =	Mean square	error.

Strip NO.	R 0/0	Strip NO.	R 0/0
1	53.6	7	53.3
2	54.5	8	54.4
3	53.5	9	54.8
4	54.7	10	53.4
5	53.6	11	54.2
6	53.5	12	53.6

Slit-width	Intensity	Slit-width	Intensity
13.8	13.5	24.2	22.7
21.75	21.7	33.4	31.2
28.3	28.1	42.5	46.2
36.2	36.7

Wave-length.	Error.	Wave-length.	Error.
0.6,e	1.1 O/o		1.9%
0.5.,	1.3%	0.3.,	3.0 o/o


1
			gt; Vj^ m
» I V 1 \ , \ , V Sr^, \ \ w-	\ J \ o- --or N r' . \ J/	y/ /d y	(
	\ or quot;	/ / .y

Wave-length /t.	Maximum %	Minimum °lo	Average %	Standard deviation	M. S. E. of average.
0.375	48.1	47.0	47.6	0.3	0.1
0.350	48.6	47.8	48.1	0.3	0.1
0.325	47.1	46.5	46.8	0.2	0.1
0.300	45.5	44.9	45.3	0.2	0.1
0.275	45.9	44.7	45.0	0.4	0.2
0.250	48.0	47.0	47.5	0.3	0.1
0.240	52.5	49.3	50.8	1.3	0.6
0.230	53.2	52.6	52.9	0.2	0.1

Wavelength M	Maximum %	Minimum %	Average %	Standard deviation	M. S. E. of Average
0.578	59.0	54.9	56.4	1.2	0.4
0.546	56.5	54.2	54.9	0.9	0.3
0.435	53.5	52.6	52.9	0.4	0.15
0.405	53.0	51.6	52.4	0.5	0.2
0.365	54.5	53.0	54.0	0.5	0.2
0.313	54.4	50.5	53.0	0.9	0.3

				!
\					o ✓
i 1 1 s o		tr' • /•	V \	or'	m
o t \ 1	\ \ \ \ V--	✓ ✓ / /
-

Wave-	R	Wave-	R	Wave-	P
length	°lo	length	R	length
in n	°lo	in ft	%	in /I	%
0.230	57.7	0.70	55.0	1.50	72.6
0.240	55.0	0.75	54.3	1.60	77,0
0.250	52.0	0.80	53.4	1.70	81.0
0.275	49.5	0.85	53.3	1.80	84.5
0.300	49.5	0.90	54.1	1.90	87.5
0.325	51.3	0.95	55.3	2.00	90.0
0.350	52.4	1.00	57,6	2.10	91.8
0.375	51.7	1.05	60.1	2.25	93.0
0.400	50.4	1.10	62.2	2.50	93.8
0.425	50.4	1.15	64.0	2.75	94.0
0.450	50.8	1.20	65.5	3.00	94.3
0.50	52.1	1.25	66.2	3.50	94.5
0.55	53.1	1.30	66.5	4.00	94.8
0.60	54.0	1.35	67.0	5.00	95.3
0.65	54.7	1.40	68.5	6.00	95.8

Temp. °K	Number of obs.	Increase %	M. S. E.
0.254,..
1600 1800 2000 2200 2400	10 10 10 10 10	3.7 4.2 4.8 5.2 5.8	0.2 0.3 0.4 0.4 0.4
0.313,t.
1600 1800 2000 2200 2400	7 7 7 7 7	2.5 2.7 3.5 4.0 4.3	0.4 0.55 0.6 0.55 0.4
0.365,1.
1600 1800 2000 2200	5 5 5 5	0.9 1.3 1.4 1.4	0.4 0.1 0.2 0.45
0.405,..
1600 1800 2000	5 5 5	2.9 f 3.4 4.0	0.5 0.4 0.4
0.435
1600 1800 2000	3 3 3	3.5 4.4 5.3	0.4 0.2 0.2

\

\
1	—iir
1


			_ »
X	/ / ___	✓ ___-	\ *
	/ /	- -

					,,0.435 ya ^ 0 254 /i
					^,-0.405 ^0 313 /i

			—-		-0.365 /I

I	Drawn	Dotted
in fl	curve	curve
0.23 fx	3.0 o/o	1.5 %
0.24 „	2.9 ..	2.0
0.25	2.8 „	2.5 „
0.275 „	2.7 „	3.6 „
0.30 ..	2.4 „	2.8 ..
values in	table 17 are	the averages of

0.254		52	2.8	0.7	0.1
0.313		31	2.1	0.7	0.1'
0.365		16	0.8	0.8	0.2
0.405		12	2.3	0.6	0.2
0.435		6	2.9 '	0.4	0.15
B.	Observations with 500 watt lamp up to 1400 °K.
0.40		4	3.1	0.7	0.35
0.45		4	3.1	0.4	0.2
0.50		4	2.8	0.5	0.25
0.55		4	2.5	0.7	0.35
0.60		4	2.3	0.2	0.1
0.65		4	1.6	0.7	0.35
C.	Observations with sodium vapor lamp up to				1600 °K.
0.58		12	2.3	0.7	0.2

in fl	dR o/o	M. S. E.	X in fl	dR %	M. S. E.
0.23	2.3	0.3	0.55	2.3	0.3
0.24	2.5	0.3	0.60	1.9	0.3
0.25	2.7	0.3	0.65	1.8	0.3
0.275	3.2	0.3	0.70	2.0	0.4
0.30	2.6	0.3	0.75	3.8	0.6
0.325	1.7	0.3	0.80	6.7	0.9
0.35	1.0	0.3	0.85	8.2	1.0
0.375	1.1	0.3	0.90	8.8	1.0
0.40	2.6	0.3	0.95	8.2	1.0
0.425	3.4	0.3	1.00	7,0	0.8
0.45	3.4	0.3	1.05	5.7	—
0.50	2.9	0.3

T =	1000	1200	1400	1600	1800	2000° K
X in /i
0.23	0.414	0.411	0.408	0.406	0.403	0.400
0.24	.440	,438	.435	.432	.429	.427
0.25	.470	.467	.465	.462	.459	.456
0.275	.494	.491	.488	.485	.482	.478
0.30	.496	.493	.491	.488	.486	.483
0.325	.481	.479	.477	.476	.474	.472
0.35	.472	.471	.470	.469	.468	.467
0.375	.479	.478	.477	.476	.475	.473
0.40	.487	.484	.482	.479	.476	.474
0.425	.484	.481	.477	.473	.470	.467
0.45	.480	.477	.473	.470	.466	.463
0.50	.468	.465	.462	.459	.456	.453
0.55	.460	.458	.456	.453	.451	.448
0.60	.453	.451	.449	.447	.445	.443
0.65	.446	.444	.442	.440	.438	.436
0.70	.442	.440	.438	.436	.434	.431
0.75	.443	.438	.434	.430	.426	.422
0.80	.441	.434	.427	.420	.412	.405
0.85	.436	.428	.419	.410	.401	.393
0.90	.426	.416	.407	.397	.388	.378
0.95	.415	.406	.397	.388	.379	.370
1.00	.396	.388	.380	.372	.364	.355
1.05	.375	.368	.361	.354	.348	.341

	2200	2400	2600	2800	3000° K
X in n
0.23	0.398	0.395	0.392	0.390	0.387
0.24	.424	.421	.418	.416	.413
0.25	.453	.450	.448	.445	.442
0.275	.475	.472	.469	.466	.463
0.30	.481	.478	.475	.473	.470
0.325	.470	.469	.467	.465	.463
0.35	.466	.465	.464	.463	.462
0.375	.472	.471	.470	.469	.468
0.40	.471	.468	.466	.463	.460
0.425	.463	.460	.457	.453	.450
0.45	.459	.456	.452	.449	.445
0.50	.450	.447	.444	.441	.438
0.55	.446	.443	.441	.439	.436
0.60	.441	.438	.436	.434	.432
0.65	.434	.432	.430	.428	.426
0.70	.429	.427	.425	.423	.420
0.75	.418	.414	.410	.405	.401
0.80	.398	.391	.384	.377	.370
0.85	.384	.375	.366	.358	.349
0.90	.369	.359	.350	.340	.331
0.95	.361	.352	.343	.334	.325
1.00	.347	.339	.331	.323	.315
1 .05	.334	.327	.320	.313	.306 1

r-gt;in °K	1200	1600	2000	2400	2800
X in n	Errors in %
0.23 to 0.65	0.6	0.6	0.7	0.8	0.9
0.7	0.7	0.9	1.0	1.3	1.6
0.8	1.2	1.8	2.3	2.7	3.5
0.9	1.5	2.1	3.0	3.8	5.0
1.0	1.3	1.9	2.6	3.5	4.5

'•1 j		0.30	0.40	0.50	0.60	0.80	1.00
0.25	ei = e2 =	1.0 1.6	1.0 3.5	1.0 4.6	1.0 ! 5.5	2.4 6.3	2.7 6.9
0.30	e\ =- 62 =		1.0 1.8	1.0 3.0	1.0 3.9	2.4 4.6	2.7 5.3
0.40	e, = e2 =			1.0 1.2	1.0 2.1	2.4 2.9	2.7 3.5
0.50	ei = ei =				1.0 0.9	2.4 1.7	1 i 2.7 2.3
0.60	ei = ei =				•	2.4 0.9	2.7 1.4
0.80	ei = ^2 =						3.5 0.6

T	T^ New	T^ Old
1200	1210	1210
1600	1616	1619
2000	2023	2033
2400	2432	2452
2800	2844	2878