James Clerk Maxwell.

The scientific papers of James Clerk Maxwell (Volume 1) online

. (page 24 of 50)
Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 24 of 50)
Font size
QR-code for this ebook


following experiments relate to the combinations of six well-defined colours only,
and I shall describe them the more minutely, as I hope to induce those who
have good eyes to subject them to the same trial of skill in distinguishing
tints.

The method of performing the experiments is described in the Transactions
of the Royal Society of Edinburgh, Vol. xxi. Part 2. The colour- top or teetotum
which I used may be had of Mr J. M. Bryson, Edinburgh, or it may be easily
extemporized. Any rotatory apparatus which will keep a disc revolving steadily
and rapidly in a good light, without noise or disturbance, and can be easily
stopped and shifted, will do as well as the contrivance of the spinning-top.

The essential part of the experiment consists in placing several discs of
coloured paper of the same size, and slit along a radius, over one another, so
that a portion of each is seen, the rest being covered by the other discs. By
sliding the discs over each other the proportion of each colour may be varied,
and by means of divisions on a circle on which the discs lie, the proportion of
each colour may be read off. My circle was divided into 100 parts.

On the top of this set of discs is placed a smaller set of concentric discs,
so that when the whole is in motion round the centre, the colour resulting from
the mixture of colours of the small discs is seen in the middle of that arising
from the laro-er discs. It is the object of the experimenter to shift the colours
till the outer and inner tints appear exactly the same, and then to read off the
proportions.

It is easy to deduce from the theory of three primary colours what must
be the number of discs exposed at one time, and how much of each colour must
appear.

Every colour placed on either circle consists of a certain proportion of each
of the primaries, and in order that the outer and inner circles may have precisely
the same resultant colour in every respect, there must be the same amount of
each of the primary colours in the outer and inner circles. Thus we have as
many conditions to fulfil as there are primary colours; and besides these we
have two more, because the whole number of divisions in either the outer or
the inner circle is 100, so that if there are three primary colours there wiU be
five conditions to fulfil, and this will require five discs to be disposable, and
these must be arranged so that three are matched against two, or four against one.
If we take six difierent colours, we may leave out any one of the six, and
so form six different combinations of five colours. It is plain that these six



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 265

combinations must be equivalent to two equations only, if the theory of three
primaries be true.

The method which I have found most convenient for registering the result
of an experiment, after an identity of tint has been obtained in the inner and
outer circles, is the following : —

Write down the names or symbols of the coloured discs each at the top of
a column, and underneath write the number of degrees of that colour observed,
calling it + when the colour is in the outer circle, and — when it is in the inner
circle ; then equate the whole to zero. In this way the account of each colour
is kept in a separate column, and the equations obtained are easily combined and
reduced, without danger of confounding the colours of which the quantities have
been measured. The following experiments were made between the 3rd and 11th
of September, 1856, about noon of each day, in a room fronting the north,
without curtains or any bright- coloured object near the window. The same
combination was never made twice in one day, and no thought was bestowed
upon the experiments except at the time of observation. Of course the gradua-
tion was never consulted, nor former experiments referred to, till each combi-
nation of colours had been fixed by the eye alone; and no reduction waa
attempted till all the experiments were concluded.

The coloured discs were cut from paper painted of the following colours : —
Vermilion, Ultramarine, Emerald-green, Snow-white, Ivory-black, and Pale
Chrome-yellow. They are denoted by the letters V, U, G, W, B, Y respectively.
These colours were chosen, because each is well distinguished from the rest, so
that a small change of its intensity in any combination can be observed. Two
discs of each colour were prepared, so that in each combination the colours might
occasionally be transposed from the outer circle to the inner.

The first equation was formed by leaving out vermilion. The remaining
colours are Ultramarine-blue, Emerald-green, White, Black, and Yellow. We
might suppose, that by mixing the blue and yellow in proper proportions, we
should get a green of the same hue as the emerald-green, but not so intense,
80 that in order to match it we should have to mix the green with white to
dilute it, and with black to make it darker. But it is not in this way that we
have to arrange the colours, for our blue and yellow produce a pinkish tint, and
never a green, so that we must add green to the combination of blue and yellow,
to produce a neutral tint, identical with a mixture of white and black.



266 EXPERIMENTS ON THE PERCEPTION OF COLOUR.

Blue, green, and yellow must therefore be combined on the large discs, and
stand on one side of the equation, and black and white, on the small discs, must
stand on the other side. In order to facilitate calculations, the colours are
always put down in the same order; but those belonging to the small discs
are marked negative. Thus, instead of writing

54U + UG + 32Y = 32W + 68B,
we write +54U + 14G-32W-68B + 32Y = 0.

The sum of all the positive terms of such an equation is 100, being the
whole number of divisions in tne circle. The sum of the negative terms is
also 100.

The second equation consists of all the colours except blue ; and in this
way we obtain six different combinations of five colours.

Each of these combinations was formed by the unassisted judgment of my
eye, on six different occasions, so that there are thirty-six independent observa-
tions of equations between five colours.

Table I. gives the actual observations, with their dates.

Table II. gives the result of summing together each group of six equations.

Each equation in Table 11. has the sums of its positive and negative co-
eflBcients each equal to 600.

Having obtained a number of observations of each combination of colours,
we have next to test the consistency of these results, since theoretically two
equations are sufficient to determine all the relations among six colours. We
must therefore, in the first place, determine the comparative accuracy of the
different sets of observations. Table III. gives the averages of the errors of
each of the six groups of observations. It appears that the combination IV. is
the least accurately observed, and that VI. is the best.

Table IV. gives the averages of the errors in the observation of each colour
in the whole series of experiments. This Table was computed in order to detect
any tendency to colour-blindness in my own eyes, which might be less accurate
in discriminating red and green, than in detecting variations of other colours.
It appears, however, that my observations of red and green were more accurate
than those of blue or yellow. White is the most easily observed, from the



EXPERIMENTS ON THE PERCEPTION OF COLOUR. 267

brilliancy of the colour, and black is liable to the greatest mistakes. I would
recommend this method of examining a series of experiments as a means of
detecting partial colour-blindness, by the different accuracy in observing differ-
ent colours. The next operation is to combine all the equations according to
their values. Each was first multiplied by a coefficient proportional to its ac-
curacy, and to the coefficient of white in that equation. The result of adding
all the equations so found is given in equation (W).

Equation (Y) is the result of similar operations with reference to the
yellow on each equation.

We have now two equations, from which to deduce six new equations, by
eliminating each of the six colours in succession. We must first combine the
equations, so as to get rid of one of the colours, and then we must divide by
the sum of the positive or negative coefficients, so as to reduce the equations
to the form of the observed equations. The results of these operations are given
in Table V., along with the means of each group of six observations. It will
be seen that the differences between the results of calculation from two equations
and the six independent observed equations are very small. The errors in red
and green are here again somewhat less than in blue and yellow, so that there
is certainly no tendency to mistake red and green more than other colours.
The average difference between the observed mean value of a colour and the
calculated value is 77 of a degree. The average error of an observation in any
group from the mean of that group was '92. No observation was attempted
to be registered nearer than one degree of the top, or yo7 of ^ circle ; so that
this set of observations agrees with the theory of three primary colours quite
as far as the observations can warrant us in our calculations ; and I think that
the human eye has seldom been subjected to so severe a test of its power of
distinguishing colours. My eyes are by no means so accurate in this respect as
many eyes I have examined, but a little practice produces great improvement
even in inaccurate observers.

I have laid down, according to Newton's method, the relative positions of
the five positive colours with which I worked. It will be seen that W lies
within the triangle VUG, and Y outside that triangle.

The first combination. Equation I., consisted of blue, yellow, and green,
taken in such proportions that their centre of gravity falls at W,



268



EXPERIMENTS ON THE PERCEPTION OF COLOUR.



In Equation II. a mixture of red and green, represented in the diagram
by the point 2, is seen to be equivalent to a mixture of white and yellow, also
represented by 2, which is a pale yellow tint.

Equation III. is between a mixture of blue and yellow and another of
white and red. The resulting tint is at the intersection of YU and WV ; that
is, at the point 3, which represents a pale pink grey.

Equation IV. is between VG and UY, that is, at 4, a dirty yellow.

Equation V. is between a mixture of white, red, and green, and a mixture
of blue and yellow at the point 5, a pale dirty yellow.

Equation VI. has W. for its resulting tint.



Blue, U.



Bed, V




G, Green.



Y, Yellow.

Of all the resulting tints, that of Equation IV. is the furthest from white ;
and we find that the observations of this equation are affected with the greatest
errors. Hence the importance of reducing the resultant tint to as nearly a
neutral colour as possible.

It is hardly necessary for me to observe, that the whole of the numerical
results which I have given apply only to the coloured papers which I used,
and to them only when illuminated by daylight from the north at mid-day in
September, latitude 55". In the evening, or in winter, or by candlelight, the
results are very different. I believe, however, that the results would differ far
less if observed by different persons, than if observed under different lights ;
for the apparatus of vision is wonderfully similar in different eyes, and even in
colour-blind eyes the system of perception is not different, but defective.



EXPERIMENTS ON THE PERCEPTION OF COLOUR.



269



Table I. — The observations arranged in groups.



Equation I.


V = 0.


+ U.


+ G.


-W.


-B. +Y.


Equation IV.


-V.


+u.


-O.


w=o.


+ B.


+ Y.


1856, Sept. 3.





54


12


34


66 34


1856,


Sept. 3.


62


15


38





53


32


4.





58


14


31


69 28




4.


63


17


37





46


37


5.





55


12


32


68 33




5.


64


16


36





50


34


6.





54


14


32


68 32




6.


62


19


38





46


35


8.





54


14


32


68 32




8.


62


19


38





47


34


9.





53


15


32


68 32




9.


63


17


37





49


34


Equation n.


-V.


u=o.


-G.


+ \V.


+ B. +Y.


Equation V.


+v.


-U.


+ G.


+w.


B = 0.


-Y.


Sept. 3.


59





41


9


71 20




Sept. 3.


56


47


28


16





53


4.


61





39


9


68 23




4.


57


50


25


18





50


5.


61





39


9


67 24




5.


66


49


24


20





51


6.


59





41


10


66 24




6.


55


47


27


18





53


8.


60





40


9


69 22




8.


54


49


26


20





51


9.


61





39


9


68 23




11.


56


50


27


17





50


Equation HI.


+v.


-u.


G = 0.


+w.


+ B. -Y.


Equation VI.


+v.


+ U.


+ G.


-W.


-B.


Y = 0.


Sept. 3.


20


56





28


52 44




Sept. 3.


38


27


35


24


76





4.


23


58





30


47 42




4.


39


27


34


24


76





5.


24


56





29


47 44




5.


40


26


34


24


76





6.


20


56





31


49 44




6.


38


28


34


24


76





8.


21


57





29


60 43




8.


39


28


33


24


76





9.


21


58





29


50 42




11.


39


27


34


23


77









Table II.— The sums of the observed equations.


















V.


U.


G.


W.




B.


Y.








Equation I.







+ 328


+ 81


-193


-


-407


+


191










II.


_


361





-239


+ 55


+ 409


+


136










III.


+


129


-341





+ 176


+ 295


_


259










IV.




376


+ 103


-224





+ 291


+


206










V.


+


334


-292


+ 157


+ 109







-


308










VI.


4


233


+ 163


+ 204


-143


-


-457












Table III. — The averages of the errors of the several equations from the means expressed in

j^ parts of a circle.



Equations.


I.


n.


m.


IV.


V.


VL


Errors.


•94


•85


1-05


117


ro8


•40



Table IV. — The averages of the errors of the several colours from the means in y^ parts of

a circle.
Colours. V. D. G. W. B, Y.

Errors. -83 -99 •SO -61 115 r09

Average error on the whole ^92.

The equations from which the reduced results were obtained were calculated as follow : —
Equation for (W)- (II) + 2 (III) + (V)-2 (I) -4 (VI).
Equation for (Y) = 2 (I) + 2 (II) - 3 (III) + 2 (IV) - 3 (V>



270 EXPERIMENTS ON THE PERCEPTION OF COLOUR.

These operations being performed, gave

V. U. G. W. B. Y.

(W) + 701 + 2282 + 1060-1474-3641 + 1072 = 0.
(Y) +2863-2761 + 1235 + 1131^ 299-2767 = 0.



From these were obtained the following results by elimination: —

Table V.



Equation

J r From (W) and (Y)
■ \ From observation






-54-1
-54-7


-13-9
-13-5


+ 32-0
+ 32-1


+ 68-0
+ 67-9


-32
-31-8


jj ( From (W) and (Y)
* ( From observation


-59-6
-60-2






-40-4
-39-8


+ 10-4
+ 9-2


+ 66-0
+ 68-2


+ 23-6
+ 22-6


,^^ f From (W) and (Y)
\ From observation


-21-7
-21-5


+ 57-4
+ 56-8






-30-2
-29-3


-48-1
-49-2


+ 42-6
+ 43-2


f From (W) and (Y)
( From observation


-62-4
-62-7


+ 18-6
+ 17-2


-37-6
-37-3






+ 45-7
+ 48-5


+ 35-7
+ 34-3


1 From (W) and (Y)
■ ( From observation


+ 55-6
+ 55-7


-49-0

-48-7


+ 25-2
+ 26-1


+ 19-2
+ 18-2






-51-0
-51-3



^T f From (W) and (Y) -397 -26-6 -337 +227 +77-3

^^•\ From observation -38-8 -27-2 -340 +28-3 +76-2

James Clerk Maxwell.
Glexlair, Jum 13, 1857.



[From The Quarterly Journal of Pure and Applied Mathematics, Vol. ii.



XVII. On the General Laws of Optical Instruments.

The optical effects of compound instruments have been generally deduced
from those of the elementary parts of which they are composed. The formulae
given in most works on Optics for calculating the effect of each spherical sur-
face are simple enough, but, when we attempt to carry on our calculations from
one of these surfaces to the next, we arrive at fractional expressions so com-
phcated as to make the subsequent steps very troublesome.

Euler (Acad. R. de Berlin, 1757, 1761. Acad. R. de Paris, 1765) has attacked
these expressions, but his investigations are not easy reading. Lagrange (Acad.
Berhn, 1778, 1803) has reduced the case to the theory of continued fractions
and so obtained general laws.

Gauss [Dioptrische Untersuchungen, Gottingen, 1841) has treated the subject
with that combination of analytical skiU with practical ability which he displays
elsewhere, and has made use of the properties of principal foci and principal
planes. An account of these researches is given by Prof. Miller in the third
volume of Taylor's Scientific Memoirs. It is also given entire in French by
M. Bravais in Liouvilles Journal for 1856, with additions by the translator.

The method of Gauss has been followed by Prof Listing in his Treatise
on the DioptHcs of the Eye (in Wagner's Handworterhuch der Physiologie) from
whom I copy these references, and by Prof Helmholtz in his Treatise vn
Physiological Optics (in Karsten's Cyclopadie).

The earliest general investigations are those of Cotes, given in Smith's
Optics, II. 76 (1738). The method there is geometrical, and perfectly general,
but proceeding from the elementary cases to the more complex by the method
of mathematical induction. Some of his modes of expression, as for instance his
measure of "apparent distance," have never come into use, although his results
may easily be expressed more intelligibly ; and indeed the whole fabric of



272 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS.

Geometrical Optics, as conceived by Cotes and laboured by Smith, has fallen
into neglect, except among the writers before named. Smith tells us that it
was with reference to these optical theorems that Newton said " If Mr Cotes
had lived we might have known something."

The investigations which I now offer are intended to show how simple and
how general the theory of instruments may be rendered, by considering the
optical effects of the entire instrument, without examining the mechanism by
which those effects are obtained. I have thus established a theory of "perfect
instruments," geometrically complete in itself, although I have also shown, that
no instrument depending on refraction and reflexion, (except the plane mirror)
can be optically perfect. The first part of this theory was conununicated to
the Philosophical Society of Cambridge, 28th April, 1856, and an abstract will
be found in the Philosophical Magazine, November, 1856. Propositions VIII.
and IX. are now added. I am not aware that the last has been proved before.

In the following propositions I propose to establish certain rules for deter-
mining, from simple data, the path of a ray of light after passing through any
optical instrument, the position of the conjugate focus of a luminous point, and
the magnitude of the image of a given object. The method which I shall use
does not require a knowledge of the internal construction of the instrument and
derives all its data from two simple experiments.

There are certain defects incident to optical instruments from which, in the
elementary theory, we suppose them to be free. A perfect instrument must
fulfil three conditions :

I. Every ray of the pencil, proceeding from a single point of the object,
must, after passing through the instrument, converge to, or diverge from, a
single point of the image. The corresponding defect, when the emergent rays
have not a common focus, has been appropriately called (by Dr Whewell)
Astigmatism.

II. If the object is a plane surface, perpendicular to the axis of the
instrument, the image of any point of it must also lie in a plane perpendicular
to the axis. When the points of the image lie in a curved surface, it is said
to have the defect of curvature.

III. The image of an object on this plane must be similar to the object,
whether its linear dimensions be altered or not; when the image is not similar
to the object, it is said to be distorted.



ox THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 273

An image free from these three defects is said to be jycrfect.

In Fig. 1, p. 285, let A^x^a^ represent a plane object perpendicular to the
axis of an instrument represented by I., then if the instrument is perfect, as
regards an object at that distance, an image A.a.p^_ will be formed by the
emergent rays, which will have the following properties :

I. Every ray, which passes through a point a^ of the object, will pass
through the corresponding point a. of the image.

II. Every point of the image will lie in a plane perpendicular to the axis.

III. The figure A.ap^ will be similar and similarly situated to the figure

Now let us assume that the instrument is also perfect as regards an object

in the plane i?i?>,y8i perpendicular to the axis through -B„ and that the image

of such an object is in the plane B^fio and similar to the object, and we
shall be able to prove the following proposition :

Prop. I. If an instrument give a perfect image of a plane object at two
different distances from the instrument, all incident rays having a common focus
will have a common focus after emergence.

Let Pj be the focus of incident rays. Let P-,a^^ be any incident ray.
Then, since every ray which passes through a^ passes through a,,, its image after
emergence, and since every ray which passes through Z;, passes through 6,, the
direction of the ray P^a^\ after emergence must be ah..

Similarly, since a^ and ySj are the images of Oj and ^i, if P^a^^^ be any
other ray, its direction after emergence will be a„fi.y

Join a, a,, h^^„ a.xL.., hfi.,; then, since the parallel planes AjCt^a^ and BJ}^,
are cut by the plane of the two rays through P^, the intersections cTiOi and
?jjSi are parallel.

Also, their images, being similarly situated, are parallel to them, therefore
a„a, is parallel to 6^j, and the lines aJj„ and a,^^ are in the same plane, and
therefore either meet in a point P^ or are parallel.

Now take a third ray through P,, not in the plane of the two former.
After emergence it must either cut both, or be parallel to them. If it cuts
both it nuist pass through the point P., and then every other ray must pass
through P., for no line can intersect three Hues, not in one plane, without
passing through their point of intersection. If not, then all the emergent rays



274 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS.

are parallel, which is a particular case of a perfect pencil. So that for every
position of the focus of incident rays, the emergent pencil is free from astig-
matism.

Prop. II. In an instrument, perfect at two different distances, the image
of any plane object perpendicular to the axis will be free from the defects of
curvature and distortion.

Through the point P, of the object draw any line P,Q, in the plane of
the object, and through P,Q, draw a plane cutting the planes A„ B, in the hnes
ttio,, h^,. These lines will be parallel to P,Q, and to each other, wherefore
also their images, a^o,, b^„ will be parallel to P,Q, and to each other, and
therefore in one plane.

Now suppose another plane drawn through P^Q, cutting the planes A, and
B, in two other lines parallel to P,Q^. These will have parallel images in the
planes A^ and B„ and the intersection of the planes passing through the two
pairs of images wiU define the line P^Q, which will be parallel to them, and
therefore to P,Q„ and will be the image of P,Q,. Therefore P^, the image
of P,Qi is parallel to it, and therefore in a plane perpendicular to the axis.
Now if all corresponding lines in any two figures be parallel, however the lines
be drawn, the figures are similar, and similarly situated.

From these two propositions it follows that an instrument giving a perfect
image at two different distances will give a perfect image at all distances. We
have now only to determine the simplest method of finding the position and
magnitude of the image, remembering that wherever two rays of a pencil inter-
sect, all other rays of the pencil must meet, and that aU parts of a plane
object have their images in the same plane, and equally magnified or diminished.

Prop. III. A ray is incident on a perfect instrument parallel to the axis,
to find its direction after emergence.

Let a J), (fig. 2) be the incident ray, A,a, one of the planes at which an
object has been ascertained to have a perfect image. A,a, that image, similar



Online LibraryJames Clerk MaxwellThe scientific papers of James Clerk Maxwell (Volume 1) → online text (page 24 of 50)