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416 ON THE THEORY OF COMPOUND COLOURS.

not merely good eyes, but a power of judging as to the exact nature of the
difference between two very pale and nearly identical tints, whether they differ
in the amount of red, green, or blue, or in brightness of illumination.

In the following paper I shall first lay down the mathematical theory of
Newton's diagram, with its relation to Young's theory of the colour-sensation.
I shall then describe the experimental method of mixing the colours of the
spectrum, and determining the wave-lengths of the colours mixed. The results
of my experiments will then be given, and the chromatic relations of the
spectrum exhibited in a system of colour-equations, in Newton's diagram, and
in three curves of intensity, as in Brewster's diagram. The differences between
the results of two observers will then be discussed, shewing on what they
depend, and in what way such differences may affect the vision of persons
othei-wise free from defects of sight.

§ III. Mathematical Theory of Newton's Diagram of Colours.

Newton's diagram is a plane figure, designed to exhibit the relations of
colours to each other.

Every point in the diagram represents a colour, simple or compound, and
we may conceive the diagram itself so painted, that every colour is found at
its corresponding point. Any colour, differing only in quantity of illumination
from one of the colours of the diagram, is referred to it as a unit, and is
measured by the ratio of the illumination of the given colour to that of the
corresponding colour in the diagram. In this way the quantity of a colour is
estimated. The resultant of mixing any two colours of the diagram is found
by dividing the line joining them inversely as the quantity of each; then, if
the sum of these quantities is unity, the resultant will have the illumination
as weU as the colour of the point so found; but if the sum of the components
is different from unity, the quantity of the resultant will be measured by the
sum of the components.

This method of determining the position of the resultant colour is mathe-
matically identical with that of finding the centre of gravity of two weights,
and placing a weight equal to their sum at the point so found. We shall
therefore speak of the resultant tint as the sum of its components placed at
their centre of gravity.

ON THE THEORY OF COMPOUND COLOURS. 41/

By compounding this resultant tint with some other colour, we may find the
position of a mixture of three colours, at the centre of gravity of its components ;
and by taking these components in different proportions, we may obtain colours
corresponding to every part of the triangle of which they are the angular points.
In this way, by taking any three colours we should be able to construct a
triangular portion of Newton's diagram by painting it with mixtures of the three
colours. Of course these mixtures must be made to correspond with optical
mixtures of light, not with mechanical mixtures of pigments.

Let us now take any colour belonging to a point of the diagram outside
this triangle. To make the centre of gravity of the three weights coincide with
this point, one or more of the weights must be made negative. This, though
following from mathematical principles, is not capable of direct physical inter-
pretation, as we cannot exhibit a negative colour.

The equation between the three selected colours, x, y, z, and the new colour
u, may in the first case be written

u = x + y-\-z (1),

05, y, % being the quantities of colour required to produce u. In the second case
suppose that z must be made negative,

u = x-^-y — z (2).

As we cannot realize the term — z as a negative colour, we transpose it to the
other side of the equation, which then becomes

u-\-z = x-\-y (3),

which may be interpreted to mean, that the resultant tint, u + z, is identical
with the resultant, x-\-y. We thus find a mixture of the new colour with one
of the selected colours, which is chromatically equivalent to a mixture of the
other two selected colours.

When the equation takes the form

u — x — y — z (4),

two of the components being negative, we must transpose them thus,

u + y-\-z = x (5),

which means that a mixture of certain proportions of the new colour and two
of the three selected, is chromatically equivalent to the third. We may thus in
all cases find the relation between any three colours and a fourth, and exhibit

418 ON THE THEORY OF COMPOUND COLOURS.

this relation in a form capable of experimental verification; and by proceeding
in this way we may map out the positions of all colours upon Newton's diagram.
Every colour in nature will then be defined by the position of the corresponding
colour in the diagram, and by the ratio of its illumination to that of the
colour in the diagram.

§ lY. Method of representing Colours by Straight Lines drawn from a Point.

To extend our ideas of the relations of colours, let us form a new geome-
trical conception by the aid of solid geometry.

Let us take as origin any point not in the plane of the diagram, and let
us draw lines through this point to the different points of the diagram; then
the direction of any of these lines will depend upon the position of the point
of the diagram through which it passes, so that we may take this line as the
representative of the corresponding colour on the diagram.

In order to indicate the quantity of this colour, let it be produced beyond
the plane of the diagram in the same ratio as the given colour exceeds in
illumination the colour on the diagram. In this way every colour in nature will
be represented by a line drawn through the origin, whose direction indicates
the quality of the colour, while its length indicates its quantity.

Let us find the resultant of two colours by this
method Let O be the origin and AB be a section
of the plane of the diagram by that of the paper.
Let OP, 0^ be lines representing colours, A, B the

OP

corresponding points in the diagram ; then the quantity of P will be jr-^ —P>

and that of Q will be jyD = 9.- The resultant of these will be represented in

the diagram by the point C, where AC : CB wq-.p, and the quantity of the
resultant will be p + q, so that if we produce OC to R, so that OR = (p-\-q)OC,
the line OR will represent the resultant of OP and OQ in direction and
magnitude. It is easy to prove, from this construction, that OR is the diagonal
of the parallelogram of which OP and OQ are two sides. It appears therefore
that if colours are represented in quantity and quality by the magnitude and
direction of straight lines, the rule for the composition of colours is identical

ON THE THEORY OF COMPOUND COLOURS. 419

witli that for the composition of forces in mechanics. This analogy has been
well brought out by Professor Grassmann in Poggendorflf's Annalen, Bd. lxxxix.

We may conceive an arrangement of actual colours in space founded upon
this construction. Suppose each of these radiating lines representing a given
colour to be itself illuminated with that colour, the brightness increasing from
zero at the origin to unity, where it cuts the plane of the diagram, and
becoming continually more intense in proportion to the distance from the origin.
In this way every colour in nature may be matched, both in quaUty and
quantity, by some point in this coloured space.

If we take any three lines through the origin as axes, we may, by co-ordi-
nates parallel to these lines, express the position of any point in space. That
point will correspond to a colour which is the resultant of the three colours
represented by the three co-ordinates.

This system of co-ordinates is an illustration of the resolution of a colour
into three components. According to the theory of Young, the human eye is
capable of three distinct primitive sensations of colour, which by their composition
in various proportions, produce the sensations of actual colour in all their varieties.
Whether any kinds of light have the power of exciting these primitive sensations
separately, has not yet been determiaed.

If colours corresponding to the three primitive sensations can be exhibited,
then all colours, whether produced by light, disease, or imagination, are com-
pounded of these, and have their places within the triangle formed by joining
the three primaries. If the colours of the pure spectrum, as laid down on the
diagram, form a triangle, the colours at the angles may correspond to the primitive
sensations. K the curve of the spectrum does not reach the angles of the circum-
scribing triangle, then no coloiir in the spectrum, and therefore no colour in
nature, corresponds to any of the three primary sensations.

The only data at present existing for determining the primary colours, are
derived from the comparison of observations of colour-equations by colour-blind,
and by normal eyes. The colour-blind equations ditfer from the others by the
non-existence of one of the elements of colour, the relation of which to known
colours can be ascertained. It appears, from observations made for me by two
colour-blind persons*, that the elementary sensation which they do not possess
is a red approaching to crimson, lying beyond both vermilion and carmine. These

♦ Trfmsactiona of the Royal Society of Edinburgh, Vol. xiL Pt 2, p. 286.

420 ON THE THEORY OF COMPOUND COLOURS.

observations are confirmed by those of Mr Pole, and by others which I have
obtained since. I have hopes of being able to procure a set of colour-blind
equations between the colours of the spectrum, which will indicate the missing
primary in a more exact manner.

The experiments which I am going to describe have for their object the
determination of the position of the colours of the spectrum upon Newton's
diagram, from actual observations of the mixtures of those colours. They were
conducted in such a way, that in every observation the judgment of the observer
was exercised upon two parts of an illuminated field, one of which was so
adjusted as to be chromatically identical with the other, which, during the whole
series of observations, remained of one constant intensity of white. In this way
the efiects of subjective colours were entirely got rid of, and all the observa-
tions were of the same kind, and therefore may claim to be equally accurate ;
which is not the case when comparisons are made between bright colours of
different kinds.

The chart of the spectrum, deduced from these observations, exhibits the
colours arranged very exactly along two sides of a triangle, the extreme red and
violet forming doubtful portions of the third side. This result greatly simplifies
the theory of colour, if it does not actually point out the three primary colours
themselves.

§ V. Description of an Instruinent for making definite Mixtures of the
Colours of the Spectrum.

The experimental method which I have used consists in forming a combi-
nation of three colours belonging to different portions of the spectrum, the quantity
of each being so adjusted that the mixture shall be white, and equal in intensity
to a given white. Fig. 1, Plate VI. p. 444, represents the instrument for
making the observations. It consists of two tubes, or long boxes, of deal, of
rectangular section, joined together at an angle of about 100".

The part AK is about five feet long, seven inches broad, and four deep ;
KN is about two feet long, five inches broad, and four deep ; BD is a partition
parallel to the side of the long box. The whole of the inside of the instrument
is painted black, and the only openings are at the end AC, and at E. At the
angle there is a Hd, which is opened when the optical parts have to be adjusted
or cleaned.

ON THE THEORY OF COMPOUND COLOURS. 421

At -£^ is a fine vertical slit ; Z is a lens ; at P there are two equilateral
prisms. The slit E, the lens L, and the prisms P are so adjusted, that when
light is admitted at -fiJ a pure spectrum \a formed at AB, the extremity of the
long box. A mirror at M is also adjusted so as to reflect the light from E
along the narrow compartment of the long box to BC. See Fig. 3.

At ^5 is placed the contrivance shewn in Fig. 2, Plate I. ^'^ is a rect-
angular frame of brass, having a rectangular aperture of 6 x 1 inches. On this
frame are placed six brass sliders, A', Y, Z. Each of these carries a knife-edge
of brass in the plane of the surface of the frame.

These six moveable knife-edges form three sUts, X, Y, Z, which may be
so adjusted as to coincide with any three portions of the pure spectrum formed
by Hght from E. The intervals behind the sliders are closed by hinged shutters,
which allow the sliders to move without letting hght pass between them.

The inner edge of the brass frame is graduated to twentieths of an inch,
so that the position of any slit can be read off. The breadth of the slit is
ascertained by means of a wedge-shaped piece of metal, six inches long, and
tapering to a point from a breadth of half an inch. This is gently inserted into
each sht, and the breadth is determined by the distance to which it enters, the
divisions on the wedge corresponding to the 200th of an inch difference in

Now suppose hght to enter at E, to pass through the lens, and to be
refracted by the two prisms at P; a pure spectrum, shewing Fraunhofer's lines,
is formed at AB, but only that part is allowed to pass which faUs on the three
slits X, Y, Z. The rest is stopped by the shutters. Suppose that the portion
faUing on X belongs to the red part of the spectrum ; then, of the white Hght
entering at E, only the red will come through the slit X. If we were to admit
red Hght at X it would be refracted to E, by the principle in Optics, that the
course of any ray may be reversed. If, instead of red light, we were to admit
white light at X, still only red Hght would come to E ; for aU other light
would be either more or less refracted, and would not reach the slit at E.
Applying the eye at the slit E, we should see the prism P uniformly illuminated
with red Hght, of the kind corresponding to the part of the spectrum which
falls on the slit X when Hght is admitted at E.

Let the sHt Y correspond to another portion of the spectrum, say the green ;
then, if white light is admitted at Y, the prism, as seen by an eye at E, will
be uniformly illuminated with green Hght; and if white Hght be admitted at X

422 ON THE THEORY OF COMPOUND COLOURS.

and Y simultaneously, tlie colour seen at E will be a compound of red and green,
the proportions depending on the breadth of the sUts and the intensity of the
Hght which enters them. The third sHt Z, enables us to combine any three kinds
of light in any given proportions, so that an eye at E shall see the face of the
prism at P uniformly illuminated with the colour resulting from the combination
of the three. The position of these three rays in the spectrum is found by
admitting the light at E, and comparing the position of the slits with the
position of the principal fixed lines ; and the breadth of the sHts is determined
by means of the wedge.

At the same time white light is admitted through BC to the mirror of black
glass at M, whence it is reflected to E, past the edge of the prism at P, so that
the eye at E sees through the lens a field consisting of two portions, separated
by the edge of the prism; that on the left hand being compounded of three
colours of the spectrum refracted by the prism, while that on the right hand is
white light reflected from the mirror. By adjusting the slits properly, these two
portions of the field may be made equal, both in colour and brightness, so that
the edge of the prism becomes almost invisible.

In making experiments, the instrument was placed on a table in a room
moderately lighted, with the end AB turned towards a large board covered with
white paper, and placed in the open air, so as to be uniformly illuminated by
the sun. In this way the thi'ee sHts and the mirror M were all illuminated
with white light of the same intensity, and all were affected in the same ratio
by any change of illumination; so that if the two halves of the field were
rendered equal when the sun was under a cloud, they were found nearly correct
when the sun again appeared. No experiments, however, were considered good
unless the sun remained uniformly bright during the whole series of experiments.

After each set of experiments light was admitted at E, and the position of
the fixed lines D and F of the spectrum was read off on the scale at AB. It
was found that after the instrument had been some time in use these positions
were invariable, shewing that the eye-hole, the prisms, and the scale might be
considered as rigidly connected.

ON THE THEORY OF COMPOUND C0L0UBJ8. 423

§ VI. Method of determining the Wave-length corresponding to any point
of the Spectrum on the Scale AB.

Two plane surfaces of glass were kept apart by two parallel strips of gold-
beaters' leaf, so as to enclose a stratum of air of nearly uniform thickness. Light
reflected from this stratum of air was admitted at E, and the spectrun formed
by it was examined at AB by means of a lens. This spectrum consists of a
large number of bright bands, separated by dark spaces at nearly uniform intervals,
these intervals, however, being considerably larger as we approach the violet end
of the spectrum.

The reason of these alternations of brightness is easily explained. By the
theory of Newton's rings, the light reflected from a stratum of air consists of
two parts, one of which has traversed a path longer than that of the other, by
an interval depending on the thickness of the stratum and the angle of incidence.
Whenever the interval of retardation is an exact multiple of a wave-length, these
two portions of light destroy each other by interference ; and when the interval
is an odd number of half wave-lengths, the resultant light is a maximum.

In the ordinary case of Newton's rings, these alternations depend upon the
varying thickness of the stratum ; while in this case a pencil of rays of different
wave-lengths, but aU experiencing the same retardation, is analysed into a spectrum,
in which the rays are arranged in order of their respective wave-lengths. Every
ray whose wave-length is an exact submultiple of the retardation will be destroyed
by interference, and its place will appear dark in the spectrum; and there will
be as many dark bands seen as there are rays whose wave-lengths ftdfil this
condition.

If, then, we observe the positions of the dark bands on the scale AB,
tlie wave-lengths corresponding to these positions will be a series of submultiples
of the retardation.

Let us call the first dark band visible on the red side of the spectrum zero,
and let us number them in order 1, 2, 3, &c. towards the violet end. Let N
be the number of undulations corresponding to the band zero which are con-
tained in the retardation R; then if n be the number of any other band, N+n
wiU be the number of the corresponding wave-lengths in the retardation, or in
symbols,

R = (N+n)\ (6).

424 ON THE THEORY OF COMPOUND COLOUBS.

Now observe the position of two of Fraunhofer's fixed lines with respect to
the dark bands, and let n„ n^ be their positions expressed in the number of
bands, whole or fractional, reckoning from zero. Let Xj, X, be the wave-lengths
of these fixed lines as determined by Fraunhofer, then

R = (N+n,)K = (N+n,)K (7);

whence N^-^^Jj^X^^n, (8),

and -R = v' _ jj KK W

Having thus found N and R, we may find the wave-length corresponding to
the dark band n from the formula

X = ^ (10).

In my experiments the line D corresponded with the seventh dark band, and
F was between the 15th and 16th, so that n^=15'7. Here then for D,

. „ „ ^'~,'rr« ^~■.►rn.r '^ Fraunhofcr's measure (11),

and for F, 7i,= 15-7, X,= 1794J "^ '

whence we find iV=34, i2 = 89175 (12).

There were 22 bands visible, corresponding to 22 different positions on the
scale AB, as determined 4th August, 1859.

Table I.

Band.

Scale.

Band.

Scale.

Band.

Scale.

n= 1

17

«= 9

36

n= 16

57

2

19

10

39

17

61

3

21i

11

42

18

65

4

23J

12

45

19

69

5

26

13

48

20

73

6

28^

U

51

21

77

7

31

15

54

22

82

8

33|

Sixteen equidistant points on the scale were chosen for standard colours
in the experiments to be described. The following Table gives the reading on
the scale AB, the value of N+n, and the calculated wave-length for each of
these : —

ON THE THEORY OF COMPOUND COLOURS. 425

Table II.

oale.

(N+«).

Wave-length.

Ck)lour.

20

36-4

2450

Red.

24

38-3

2328

Scarlet

28

39-8

2240

Orange.

32

41-4

2154

Yellow.

36

42-9

2078

Yellow-Green.

40

44-3

2013

Green.

44

45^7

1951

Green.

48

47-0

1879

Bluish green.

52

48-3

1846

Blue-green.

56

49-6

1797

Greenish blue.

60

50-8

1755

Blue.

64

51-8

1721

Blue.

68

52-8

1688

Blue.

72

53-7

1660

Indigo.

76

54-7

1630

Indigo.

80

55-6

1604

Indiga

Having thus selected sixteen distinct points of the spectrum on which to
operate, and determined their wave-lengths and apparent colours, I proceeded
to ascertain the mathematical relations between these colours in order to lay
them down on Newton's diagram. For this purpose I selected three of these
as points of reference, namely, those at 24, 44, and 68 of the scale. I chose
these points because they are weU separated from each other on the scale, and
because the colour of the spectrum at these points does not appear to the eye
to vary very rapidly, either in hue or brightness, in passing from one point to
another. Hence a small error of position will not make so serious an alteration
of colour at these points, as if we had taken them at places of rapid variation ;
and we may regard the amount of the illumination produced by the light
entering through the slits in these positions as sensibly proportional to the

(24) corresponds to a bright scarlet about one-third of the distance from
C to D; (44) is a green very near the line E; and (68) is a blue about one-
third of the distance from F to G.

42(3 ON THE THEORY OF COMPOUND COLOURS.

§ VII. Method of Observation.

The instrument is turned with the end AB towards a board, covered with
white paper, and illuminated by sunlight. The operator sits at the end AB, to
move the sliders, and adjust the sHts ; and the observer sits at the end E,
which is shaded from any bright light. The operator then places the sHts so
that their centres correspond to the three standard colours, and adjusts their
breadths till the observer sees the prism iQuminated with pure white light of
the same intensity with that reflected by the mirror M. In order to do this,
the observer must tell the operator what difference he observes in the two halves
of the illuminated field, and the operator must alter the breadth of the slits
accordingly, always keeping the centre of each sKt at the proper point of the
scale. The observer may call for more or less red, blue or green; and then
the operator must increase or diminish the width of the slits X, Y, and Z
respectively. If the variable field is darker or lighter than the constant field,
the operator must Aviden or narrow all the slits in the same proportion. When
the variable part of the field is nearly adjusted, it often happens that the
constant white light from the mirror appears tinged with the complementary
colour. This is an indication of what is required to make the resemblance of
the two parts of the field of view perfect. When no difference can be detected
between the two parts of the field, either in colour or in brightness, the observer
must look away for some time, to relieve the strain on the eye, and then look
again. If the eye thus refreshed still judges the two parts of the field to be
equal, the observation may be considered complete, and the operator must measure
the breadth of each slit by means of the wedge, as before described, and write
down the result as a colour-equation, thus —
Oct. 18, J. 18-5 (24) + 27 (44) + 37 (68) = W-^'^ (13).

This equation means that on the 18th of October the observer J. (myself) made
an observation in which the breadth of the slit X was 18-5, as measured by
the wedge, while its centre was at the division (24) of the scale ; that the breadths

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