PART II. CONTRIBUTIONS TO SCIENCE.

“He was one of those who took more delight in the contemplation of truth than in the praise of having discovered it.”

Playfair’s Memoirs of Dr. Hutton.

AT the close of the memoir published in the Proceedings of the Royal Society, Mr. W. D. Niven, referring to Maxwell’s scientific work, says:—

It is seldom that the faculties of invention and exposition, the attachment to physical science, and capability of developing it mathematically, have been found existing in one mind to the same degree. It would, however, require powers somewhat akin to Maxwell’s own, to describe the more delicate features of the works resulting from this combination, every one of which is stamped with the subtle but unmistakable impress of genius.

It will probably be many years before an approximate estimate can be formed of the value of Maxwell’s work. In the following pages no attempt has been made to give more than a brief account of a few of his principal contributions to science. The chief subjects referred to have, for convenience, been arranged in the following order:—

1. Experiments on Colour Vision and other contributions to Optics.

2. Investigations respecting Elastic Solids.

3. Pure Geometry.

4. Mechanics.

5. Saturn’s Rings.

6. Faraday’s Lines of Force, and Maxwell’s Theory of the Electro-magnetic Field, including the Electro-magnetic Theory of Light and other investigations in Electricity.

7. Molecular Physics.

1. The subject of colour-vision attracted Clerk Maxwell’s attention at an early period. In dealing with phenomena of this class, we must remember that it is necessary to distinguish between the sensation itself and its physical cause, or between the subjective and objective aspects of the same phenomenon. Thus a pure musical tone consists of a regular succession of similar vibrations, and two tones may differ in the extent of these vibrations, and in the number which take place in a second. Corresponding to these physical differences, are experienced differences in the intensity or loudness, and the pitch of the note. Light, like sound, consists objectively of certain periodic disturbances or vibrations of a medium, but differs from sound in the character of the motion, the nature of the medium which transmits it, and the number of vibrations which take place in a second. The simplest kind of light consists, like a pure tone in sound, of a regular succession of similar vibrations, the extent of which determines the intensity of the light, while their rapidity corresponds to the colour sensation produced. The constitution of ordinary white light is much more complicated. Newton allowed a beam of sunlight to pass through a prism, and then to fall upon a screen, when, instead of a white patch of light, he obtained a spectrum, or continuous band of colour, varying from crimson through scarlet, orange, yellow, green, blue, to violet. Rays corresponding to this infinite variety of colour must, therefore, exist together in white solar light, and these rays differ physically from one another in the rapidity of the vibrations of which they consist; the deep crimson corresponding to less than 400,000,000,000,000 vibrations per second, and the extreme violet to more than 700,000,000,000,000, the length of a wave in air being, in the first case, about of an inch, and, in the second case, about of an inch.

But white light is not necessarily of so complex a nature as sunlight; thus it may consist of a mixture of red, green, and blue lights simply, or of yellow and blue, or greenish-yellow and violet, or of other mixtures of two or more kinds of light, each in itself homogeneous. When this is the case, the constitution of the light is revealed when it is allowed to pass through a prism, for it is then separated into its simple constituents which, on emerging from the prism, pursue different paths. Similarly, various kinds of homogeneous light may be matched by means of a mixture of lights of other colours appropriately chosen. Thus orange light of a particular hue may be the homogeneous light of the spectrum, or it may be a mixture of red and yellow or of red and green lights, which is chromatically identical with the homogeneous orange, but optically different, inasmuch as the mixture can be resolved into its constituents by means of a prism, while the homogeneous orange light may be passed through any number of prisms and yet retain its perfect homogeneity.

These results were accounted for by Thomas Young (1801), on the supposition that there are three separate sensations which are excited to different degrees—the proportion in which each is excited depending on the nature of the light. He was of opinion that these three sensations were red, green, and violet, and that all other hues were compound colours, though they might correspond to a simple kind of light. “The quality of any colour depends, according to this theory, on the ratio of the intensities of the three sensations which it excites, and its brightness depends on the sum of these three intensities.”

Young’s colour diagram was a triangle, at the angular points of which he placed the three colours corresponding, according to his theory, with the primary sensations; along the sides of the triangle were placed the colours formed by mixing these two and two together, while within the triangle were to be found the colours resulting from mixtures of all three in different proportions. The position of each colour was determined by finding the centre of gravity, of two or three heavy particles placed at the angular points of the triangle, the weight of each particle being proportional to the quantity of the corresponding light employed in the mixture.

Although the light of the spectrum which corresponds to any particular spectral colour is simple in its constitution, consisting of a definite number of waves per second, and ordinary white light is a combination of an infinite number of such rays, varying in rapidity of vibration from the extreme red to the extreme violet, yet the mechanism of vision appears to be of a threefold character, so organised that light of each particular wave-length affects in different proportions the three colour senses, the hue depending on the relative amounts to which they are severally excited, as explained by Young. An analogy may help to make this distinction, clearer; thus, it has been customary to speak of the heating power, the illuminating power, and the photographic power, of any particular kind of light; and in the solar spectrum formed by a glass prism the greatest heating power is possessed by the ultra red rays, the greatest illuminating power (judged by normal eyes) by the greenish yellow rays, and the greatest photographic power (in relation to silver salts) by the ultra-violet rays; but we do not thereby imply that there are three separate classes of rays in solar light, consisting respectively of heating rays, illuminating rays, and chemical rays: we simply mean that any particular homogeneous ray possesses each of these three powers to a certain extent. Similarly any ray of the spectrum may be capable of stimulating each of three independent colour sensations, but it does not follow that it consists physically of three different kinds of light.

In the Philosophical Magazine for 1852 there is an account of an experiment by Helmholtz, which consisted in mixing the colours of the spectrum by forming two spectra from slits at right angles to each other. It is obvious that if the breadth of each spectrum be equal to its length, every pair of colours in the whole spectrum will in this way be superposed. In Helmholtz’s experiment it appeared that a mixture of yellow and indigo produced white.

Maxwell’s first published experiments on the mixture of coloured lights, executed while he was a B.A. at Cambridge, had reference to the mixtures of coloured lights obtained from coloured papers. One of the best accounts of these experiments will be found in the paper read before the Royal Society of Edinburgh by Clerk Maxwell on 19th March 1855.

Fig. 1.

The instrument first employed was a top, constructed by Maxwell himself; but he afterwards had tops made by Mr. Bryson of Edinburgh. The coloured papers were cut into circular discs, with a small hole in the centre to admit the spindle of the top, and were slit along a radius from the circumference to the central hole. By this means two or more discs could be placed together, and by turning one relatively to the others, more or less of any particular disc could be made visible from above. The top consisted of a disc of metal, covered with white paper and mounted on a suitable spindle, the circumference of which was divided into 100 equal parts. Two sizes of discs were employed; the larger discs having been adjusted as required, were placed on the graduated plate, and the smaller discs above them. The plate then presented an appearance similar to that sketched in the adjoining figure. The employment of the apparatus depends on the fact, observed by Hartley the psychologist, that visual impressions remain for some time on the retina even after the source of light has been removed. Thus, suppose that the larger discs are respectively red, green, and blue, and suppose that we look at the middle of the red sector and then spin the top. If the spinning is sufficiently rapid the impression produced by the red light will not have diminished sensibly before the green paper takes the place of the red, and the green light produces its effect on the same portion of the retina, and this again is followed by blue before either the red or the green sensations have sensibly diminished, and then the whole process is repeated many times in a second. The impressions due to the three coloured lights are therefore blended on the retina, and it can be shown experimentally that the tint observed is the same as when the three kinds of light in the same relative proportions are allowed to fall simultaneously upon the eye. The colour top, therefore, enables us to mix together, in any definite proportions, which can be changed at will, the lights from differently coloured papers, and to observe the effect produced by the mixing.

Suppose that the colours of the larger discs are red, green, and blue, and that these are arranged so that the coloured lights may be mixed in the proportions necessary to form white light. Then the colour of the ring exhibited by the larger discs when the top revolves can differ from that of white paper only in respect of illumination. If, therefore, we can diminish the apparent brightness of the central circle at will, we may produce a complete match. This may be effected by combining a black disc, which emits very little light indeed, with the white disc, so that the amount of light received from the central circle is approximately proportional to the angle of the white sector. On spinning the top, the central circle appears of a neutral gray, which is the same as a dull white, and the darkness of the gray increases with the amount of the black sector introduced. If, now, the outer circle appears yellow compared with the central gray when the top is spinning, it is found that, by increasing the amount of blue and diminishing the red and green, the circles may be matched in hue. Similarly, if it appears too green we must diminish the green and increase the red and blue, while if it appears purple we must increase the amount of green. After a little experience it is easy to recognise the character of the change required.

Now, suppose that we have three and only three colour sensations, and that, for the sake of argument, these correspond respectively to red, green, and blue. Then if two colours contain the same amount of red, the same amount of green, and the same amount of blue, they must be in every respect identical. There are, therefore, three conditions to satisfy in order that two colours may match. But in the case of the top, two other conditions must be also fulfilled, for the portions of the smaller discs employed must exactly fill up the circle, and the same must be true of the larger discs. Hence there are in all five conditions to be fulfilled in order to make a colour match, and in general five discs will have to be employed and matched, two against three or three against two. If there had been four primary colour sensations, we should have required in general to employ six discs to ensure a match, and so on for any other number. Maxwell showed that with a normal eye a match could always be obtained with five discs, but that with a colour-blind person a match could always be obtained with four discs, thus demonstrating that the normal eye possesses three independent colour sensations, while the eye of the colour-blind possesses only two. This conclusion will appear more evident from the experiments with the colour box, which will be described presently.

The result of each experiment was first expressed by an equation in which the colours in the outer circle, with their respective quantities as coefficients, appeared on the left of the equation, and those in the inner circle on the right side. Thus, employing for vermilion, for ultramarine, and for emerald green, for snow-white, and for black, the equation means that a sector of vermilion occupying 37 divisions of the outer circle, combined with a sector of ultramarine occupying 27 divisions, and a sector of emerald green occupying 36 divisions, produced, on spinning, a colour which matched that obtained by a sector of white paper occupying 28 divisons, and a sector of black occupying 72 divisions. The sum of the coefficients on each side of the equation is of course 1.

The mode of conducting an experiment is best described in Maxwell’s own words:—

As an example of the method of experimenting, let us endeavour to form a neutral gray by a combination of vermilion, ultramarine, and emerald green. The most perfect results are obtained by two persons acting in concert, when the operator arranges the colours and spins the top, leaving the eye of the observer free from the distracting effect of the bright colours of the papers when at rest.

After placing discs of these three colours on the circular plate of the top, and smaller discs of white and black above them, the operator must spin the top and demand the opinion of the observer respecting the relation of the outer ring to the inner circle. He will be told that the outer circle is too red, too blue, or too green, as the case may be, and that the inner one is too light or too dark as compared with the outer. The arrangements must then be changed so as to render the outer and inner circles more nearly alike. Sometimes the observer will see the inner circle tinted with the complementary colour of the outer, one. In this case the observer must interpret the observation with respect to the outer circle, as the inner circle contains only black and white.

By a little experience the operator will learn how to put his questions, and how to interpret their answers.

Plate 1. DIAGRAM SHEWING THE RELATIONS OF LIGHT.

The observer should not look at the coloured papers, nor be told the proportions of the colours during the experiments. When these adjustments have been properly made, the resultant tints of the outer and inner circles ought to be perfectly indistinguishable when the top has a sufficient velocity of rotation. The number of divisions occupied by the different colours must then be read off on the edge of the plate, and registered in the form of an equation. Thus in the preceding experiment we have vermilion, ultramarine, and emerald green outside, and black and white inside. The numbers, as given by an experiment on the 6th March 1855, in daylight without sun, are—

In a similar way matches were obtained in which the resulting tint was a decided colour and not a neutral gray. Thus on the 5th of March 1855 a match was obtained as follows— where represents pale chrome. The resulting tint in this case is an impure yellow.

Mixtures which appear to make perfect matches by one kind of light are far from matching one another when viewed by a different light. Thus, representing carmine, the following match was obtained by daylight, viz.— while by gaslight the match was which shows that the yellowing effect of the gaslight tells more on the white than on the combination of colours.”

Maxwell, following Young, represented the results of his experiments graphically by means of a triangle, at the angular points of which he placed the three colours which he believed to most nearly correspond with the primary sensations. (See Plate I.) The colours he selected for these positions were vermilion, ultramarine, and emerald green, of such strength that, when mixed in equal proportions, they produced a neutral gray which therefore appeared at the centre of the triangle. In the paper published in the Edinburgh Transactions the reason alleged for selecting green in preference to yellow is the fact that it was found possible to produce a distinct yellow from a mixture of emerald green and vermilion, but impossible to produce a green from a mixture of blue and yellow. Any colour which can be produced by a mixture of all the three primary tints will lie within the triangle; but a colour which must be mixed with one of the primary tints in order to match a mixture of the other two will lie without the triangle. Thus the pale chrome referred to above must be mixed with blue in order to match a mixture of red and green, and it must therefore be placed in such a position that the centre of gravity of the yellow and blue taken in proper proportions may be on the line joining the red and green, and may divide this line in the inverse ratio of the amounts of red and green respectively required. A copy of Maxwell’s diagram, showing the chromatic relations of coloured papers, is given in the accompanying plate (Plate II.) The original is in the Cavendish Laboratory.

As above mentioned, Maxwell found that in the case of colour-blind persons only four discs (including black) were required instead of five to ensure a match. The following is an example of a colour-blind equation:— where , , and represent green, blue, and red respectively. To a normal eye the outer circle represented by the left-hand side of the equation appears of a dark blue green, but to the colour blind this matches the full red of the smaller circle.

By comparing different colour matches, it is possible to determine the position on the colour diagram of the tint corresponding to the missing sensation in the colour-blind. Thus, if two colours appear to a colour-blind person to match, they can only differ in their effect upon a normal eye by the degree in which they excite the sensation which is missing in the colour-blind, and they must therefore lie upon a line passing through the position of this missing sensation. By determining two such lines, the position of the pure sensation is located.

Plate II. DIAGRAM ILLUSTRATING THE CHROMATIC RELATIONS OF COLOURED PAPERS.

A line drawn through this point and the position of white is, to a colour-blind person, a neutral line for its whole length (see Plate II.) The position corresponding to the pure sensation which is missing in the colour-blind is near the red, but outside the triangle, and hence corresponds to a purer red than any in the solar spectrum (see p. 480).

The colour-blind generally class all tints as yellows or blues. According to Maxwell’s theory, their sensations correspond to green and blue; but the reason why they regard yellow as brighter than green lies in the fact that although to the normal eye the sensation of yellow is a combination of the sensations of red and green, yet yellows are so much brighter than greens that light from them generally excites the green sensation more powerfully than that from green objects themselves, and hence yellows are more conspicuous than greens to the colour-blind, but only in virtue of the green they contain.

To enable colour-blind persons to distinguish between red and green, Maxwell had a pair of spectacles constructed, one eye-glass of which was red and the other green, so that the object appeared differently to the two eyes. These spectacles cause objects to appear to possess a metallic lustre on account of the different ways in which they are presented to the eyes. To the colour-blind a red object would appear brighter when seen through the red glass, while a green object would appear brighter through the green glass.

There are colour-blind persons whose vision is very different from that described by Maxwell in his early papers, but to these we shall again refer presently (p. 482).

It would occupy too much space to describe the different methods adopted by Maxwell for the purpose of mixing light of different colours, but his colour box demands a special notice, both on account of the perfection with which it is adapted to the object in view and the extreme beauty of its arrangement. The first two boxes constructed were of inconveniently large dimensions. The form we shall describe is that finally adopted, and the shell, as well as the principal optical apparatus of the box, are still at the Cavendish Laboratory. The description is taken from Maxwell's paper in the Phil. Trans. for 1860.

Fig. 2.

Fig. 2 represents the instrument. At is placed the apparatus represented in Fig. 3, in which represents a rectangular frame of brass, having a rectangular aperture of inches.

On this frame are placed six brass sliders, . Each of these carries a knife-edge of brass in the plane of the surface of the frame.

Fig. 3.

At (Fig. 2) is a fine vertical slit. At , and are three plane mirrors, each inclined at to the sides of the box. are two prisms (angles ), and is a concave silvered glass of radius, 34 inches.

If light enter the box at , it will be reflected at , and a portion of the pencil, after passing through the prisms, will be reflected from , and after again traversing the prisms will form a spectrum at .

The six movable knife edges above referred to form three slits, , which may be so adjusted as to coincide with any three portions of the pure spectrum formed by light from . The intervals between the sliders are closed with hinged shutters, which allow the sliders to move without letting light 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, 6 inches long, and tapering to a point from a breadth of half an inch. This is gently inserted into each slit, 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 breadth, so that the unit of breadth is ·005 inch. The gauge is balanced on one finger, and inserted into the slit till the pressure just causes it to slide on the finger. This (on the same principle as in Whitworth’s Millionth Measurer) ensures uniformity in the pressure in different measurements.

Now suppose light to enter at , and to be refracted by the two prisms and , and after reflection at , to again pass through the prisms. A pure spectrum, showing Fraunhofer’s lines, is formed at , but only that part is allowed to pass which falls on the three slits . The rest is stopped by the shutters. Suppose that the portion falling on belongs to the red part of the spectrum: then, of the white light entering at , only the red will come through the slit . If we were to admit red light at , it would be refracted to 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 , still only red light would come to , for all other light would be refracted more or less than the red, and would not reach the slit at . Applying the eye at the slit , we should see the prism uniformly illuminated with red light of the kind corresponding to the part of the spectrum which falls on the slit when white light is admitted at .

Let the slit correspond to another portion of the spectrum, say the green: then, if white light be admitted at , the prism, as seen by an eye at , will be uniformly illuminated with green light; and if white light be admitted at and simultaneously, the colour seen at will be a compound of red and green, the proportions depending on the breadth of the slits and the intensity of the light which enters them. The third slit enables us to add a third colour, and thus to combine any three kinds of light in any given proportions, so that an eye at shall see the face of the prism at 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 , and comparing the position of the slits with the position of the principal fixed lines.

At the same time, another portion of the light from the source (generally a sheet of white paper) enters the instrument at , is reflected at the mirror , passes through the lens , is reflected at the mirror , passes close to the edge of the prism , and is reflected along with the coloured light at , to the eye-slit at .

In this way the compound colour is compared with a constant white light in optical juxtaposition with it. The mirror is made of silvered glass, that at is made of glass roughened and blackened at the back, to reduce the intensity of the constant light to a convenient value for the experiments. By adjusting the slits properly the two portions of the field may be made equal both in colour and brightness, so that the edge of the prism becomes almost invisible. When light enters at , the instrument gives a spectrum in which Fraunhofer’s lines are very distinct, and the length of the spectrum between Fraunhofer’s lines and is 3·6 inches. The outside measure of the box is 3 feet 6 inches, by 11 inches, by 4 inches, and it can be carried about, and set up in any position, without readjustment.

In making an observation the centres of the slits are placed opposite the divisions of the scale corresponding to the colours to be mixed, the breadth of each slit being varied till the resultant light cannot be distinguished from the white light reflected from and . If the mixture differs from the standard white light only by being too bright or too dull, all the slits must be opened or closed in the same proportion. The result of any observation is expressed by an equation in which each particular colour employed is represented by the corresponding division of the scale placed in brackets, while the breadth of the slit is written as the coefficient of this colour. Thus, the equation means that the breadth of the slit was 18·5, as measured by the wedge, while its centre was at the division (24) of the scale; that the breadths of and were 27 and 37, and their positions (44) and (68); and that the illumination produced by these slits was exactly equal, in the estimation of the observer, to the constant white .

In the first instance, recorded in the paper above referred to, the observations were made by Professor Maxwell himself (J), and by Mrs. Maxwell (K). Professor Maxwell’s complexion was dark and his hair black; Mrs. Maxwell was extremely fair. The observations of each exhibited very small errors from the mean, far less than would have been expected, but there was always a difference apparent between the two observers, and this always in the same direction. A difference of this character was found by Maxwell to be general between dark and fair persons, especially when the light fell on the centre of the retina (the fovea centralis or macula lutea). To this point reference will again be made.

By considering the errors in the several, observations between the standard colours, Maxwell showed that greater accuracy is attainable in the case of red light than in that of green or blue, and that variations in colour are more easily detected than variations in brightness.

One result of the observations is especially worthy of notice. It follows from a comparison of the equations obtained in the several parts of the spectrum that any tint in the spectrum lying between the divisions (24) and (46) of the scale could be exactly imitated by mixing in proper proportions the red of (24) and the green of (46), while every tint lying between (48) and (64) could be produced by a proper mixture of the green of (48) and the blue of (64). Hence if a colour diagram be constructed as explained on page 473, the whole of the spectrum between the red of (24) and the green of (46) will lie in a straight line, forming one side of the triangle, while that lying between (48) and (64) forms another side (the included angle being not quite closed). It was also found that the violet beyond (68) lay very nearly on the line joining (68) and (24), so that the spectrum exhibited in this diagram “forms two sides of a triangle, with doubtful fragments of the third side.” Thus “all the colours of the spectrum may be compounded of those which lie at the angles of this triangle.” A diminished copy of one of Maxwell's lecture diagrams, painted with his own hand and deduced (it is believed) from experiments with the colour top, is given in Plate I.[246]

By studying Plate I., it will be seen that a mixture of yellow and blue lights produces varying shades of dirty yellow and pink, as the relative amount of blue is increased, and by properly selecting the yellow and blue a neutral gray (or white) may be obtained.

The pink tint produced by mixing blue and yellow lights had been previously noticed by Helmholtz and others. The well-known fact that an admixture of blue and yellow pigments produces generally a green was explained by Helmholtz on other grounds. Suppose, for instance, that we are provided with properly chosen blue and yellow glasses: let sunlight shine through the blue glass and fall upon a screen, and allow a second beam of sunlight, after reflection at a mirror, to pass through the yellow glass and fall on the same portion of the screen as the blue light. If the two colours are properly selected, the illuminated portion of the screen will appear white or pink. Now place the pieces of glass together, so that the same light passes through both, and the light transmitted will be green. This is explained if we examine with the prism the light transmitted by each glass. That which has passed through the blue glass will show, in addition to blue light, a certain amount of violet and green, while that which has passed through the yellow glass will comprehend, in addition to yellow light, a certain quantity of orange (perhaps red) and green. Now the only light capable of passing through both glasses is green, and this, therefore, is the colour observed on looking through the two together. Similarly, when blue and yellow pigments are ground together, the light penetrates a little way into the substance of the pigment, is reflected, and emerges, some of its constituents being absorbed by one pigment and some by the other, and the only light which both pigments transmit is green, which is consequently the colour of the emergent light, and therefore the hue of the mixture.

The set of colour equations given above, and the diagram of colours there described, were obtained from the observations of Mrs. Maxwell (K). The diagram obtained from Professor Maxwell’s (J’s) own observations was slightly different, as before mentioned.[247] The chief differences were that (J) saw more green in the orange and yellow portions of the spectrum than (K), so that in the diagram obtained from his observations these colours appeared nearer the green than in the other, while the colours between green and blue appeared more blue to (J) than to (K), and were therefore placed higher up in his diagram. Thus, when the instrument was adjusted to suit (K), one of the selected colours being (32), (36), or (40), then (J) saw the mixture too green; but if (48), (52), (56), or (60) were the selected colour when adjusted for (K), it appeared too blue to (J), “showing that there was a real difference in the eyes of these two individuals, producing constant and measurable differences in the apparent colour of objects.”

With those colour-blind persons whom Maxwell first examined only two slits were required to produce a colour chromatically identical with white; while the spectrum, a little on the red side of the line F, appeared to be identical with white. “From this point to the more refrangible end the spectrum appears to them ‘blue.' The colours on the less refrangible side appear all of the same quality, but of different degrees of brightness; and when any of them are made sufficiently bright they are called ‘yellow.'” Thus a colour-blind or dichromic person, in speaking of red, green, orange, and brown, refers to different degrees of brightness or purity of a single colour, and not to different colours. This colour he calls yellow.

Of the three standard colours the red appears to them “yellow,” but so feeble that there is not enough in the whole red division of the spectrum to form an equivalent to make up the standard white. The green at appears a good “yellow,” and the blue at from towards appears a good “blue.”

It was for these researches that Maxwell received the Rumford Medal of the Royal Society in 1860.

The only cases of colour-blindness which Maxwell met with in the course of his earlier experiments were those of persons blind to red rays. This, however, is not the only kind of colour-blindness known. Some appear to be deficient in the blue (or violet) sensation. To these persons bright yellow appears white, and the neutral line is a line drawn through the yellow and the point corresponding to the pure blue (or violet) sensation. In a valuable paper published in the Proceedings of the Royal Society, vol. XXXI. p. 302, M. Frithiof Holmgren, Professor of Physiology in the University of Upsala, gave an account of a case of red blindness, and a case of violet-blindness in each of which only one eye was really colour-blind, the vision of the other being nearly normal. This peculiarity enabled the observer to compare the experience of colours gained through one eye with that gained by the other, and thus to state in the language of those possessing normal vision, how colours really appear to the colour-blind. His results agree in their general character with Maxwell’s, but Holmgren (with many others) makes the third sensation correspond to violet instead of blue, while, according to him, the violet-blind class colours as red and yellow instead of red and green.

Though the largest and most important, Maxwell’s theory of compound colours was by no means his only, or even his earliest, contribution to Optics. While a bachelor-scholar at Trinity, at the request of Messrs. Macmillan he wrote a considerable portion of a text-book on geometrical optics. The work was never finished, but the MS. is still extant. The mode of treating the subject, as stated by Professor Maxwell in one of his letters to his father,[248] is decidedly novel, and calculated to bring down a storm of abuse upon the author. He starts by postulating the possibility of obtaining perfect images, and then investigates the laws of reflection and refraction necessary for this.

Perhaps the most remarkable of Maxwell’s contributions to Optics was his identification of the velocity of light with the ratio of two quantities, each of which is capable of being measured electrically; but this subject will be again referred to in describing his electrical researches. A paper on a general theory of optical instruments was published by Maxwell in the Quarterly Journal of Mathematics in 1858, and another on the same subject in the Proceedings of the Cambridge Philosophical Society, 1866. A paper on the best mode of projecting a spectrum on a screen appeared in the Proceedings of the Royal Society of Edinburgh in 1869. We may also refer to the papers on “The Focal Lines of a refracted Pencil,” in the Proceedings of the Mathematical Society of London for 1871-3; on a “Bow seen on the surface of ice,” in the Proceedings of the Royal Society of Edinburgh for 1872; on “The Spectra of Polarised Light,” the subject which had interested him so much in 1847 (see Part I., p. 84), in the Transactions of the Royal Society of Edinburgh for 1872; and on “Double Refraction in a Viscous Fluid in Motion,” in the Proceedings of the Royal Society for 1873, and the Ann. de Phys. and Chimie, 1874.

Among the many optical contrivances designed by Professor Maxwell, we ought not to omit to mention the real-image stereoscope. This instrument, in which ordinary stereoscopic slides are employed, consists essentially of two convex lenses of short focal length, say 4 inches, placed side by side in a wooden frame at a distance from the pictures, equal to twice their focal length, while the distance between the centres of the lenses is half the distance between the centres of the pictures. The result of this arrangement is that real images of the two pictures of the same size as the pictures themselves, formed one by one lens, and the other by the other, are superposed on the axis of the instrument at a distance in front of the lenses equal to twice their focal length, that is, as far in front of the lenses as the pictures are behind them. The double, or stereographic image so formed is viewed through a large lens at a suitable distance in front of it, the observer standing at a distance of three or four feet. Many persons[249] who cannot appreciate the ordinary box stereoscopes obtain very satisfactory effects with this instrument.

Professor Maxwell prepared a large number of stereoscope slides of geometrical figures. Among them may be mentioned the surface of centres of an ellipsoid, lines of curvature on an ellipsoid, and on elliptic and hyperbolic paraboloids, the parabolic cyclide, the horned cyclide, and the spindle cyclide, a twisted cubic with three asymptotes, a Gordian knot, etc. The wood blocks from which these slides were printed are now in the Cavendish Laboratory.

Another extremely pretty optical toy of his construction, at present in the possession of Mrs. Maxwell, is a Zoëtrope, or “Wheel of Life.” In the ordinary instrument, on looking through the slits in the revolving cylinder the figures are seen moving on the opposite side of the cylinder. Maxwell inserted concave lenses in place of the slits, the lenses being of such focal length that the virtual image of the object at the opposite extremity of the diameter of the cylinder was formed on the axis of the cylinder, and consequently appeared stationary as the cylinder revolved.[250]

In ordinary light the vibrations take place in all directions at right angles to that in which the light is being transmitted. A beam of ordinary light is therefore symmetrical on all sides. But it is possible in various ways to confine all the vibrations to one plane. Thus, if a ray be reflected from the surface of polished glass at a particular angle, all the vibrations take place in one plane, and it is generally believed that this plane is parallel to the surface of the glass, and therefore perpendicular to the plane of incidence. Since all the vibrations are now taking place in one plane, the beam of light has acquired, as it were, sides or poles, and is said to be plane polarised. There are other contrivances by which the movements may be made all to take place in regular succession in a circle about the direction of the ray as axis, and the light is then said to be circularly polarised. This may be effected by passing a beam of plane polarised light through a sheet of mica (or some other crystals) of a particular thickness.

The velocity with which waves are transmitted through a substance depends not only on the density but on the elasticity of the substance in the direction in which the vibrations take place. How, many crystals have different elasticities in different directions, and a similar condition may be induced by mechanical means in other bodies. In such substances the velocity of light depends on the direction in which the vibrations take place; and, generally, if a beam of ordinary light fall upon such a crystal it will be separated into two, one of which will consist of vibrations in the direction in which the elasticity is greatest (of all the directions which it can select), and the other in that in which the elasticity is least. These rays, being transmitted with different velocities, will be differently refracted, and part company in the crystal; and since in each ray the vibrations are in one direction only, each ray will be plane polarised, the vibrations taking place in the two rays in planes perpendicular to one another.

A Nicol’s prism consists of a long prism of Iceland spar cut into two along a diagonal plane, the segments being cemented together again by Canada balsam. The effect of the balsam is to reflect one of the polarised rays out at the side of the prism while it allows the other to pass through. Consequently, if common light fall on a Nicol’s prism, only half of it will pass through, and this will be plane-polarised in a particular plane. If another prism be placed in the path of the ray and be situated similarly to the first, the light will pass through it, but if it be turned around its axis through a right angle, the light falling on it corresponds to that which is reflected out of the prism by the balsam, and no light passes through the second prism. Under these circumstances the Nicols are said to be crossed. Such an arrangement of two Nicol’s prisms or any equivalent arrangement which may be made with glass reflectors or crystals of tourmaline, constitutes a polariscope; the first Nicol is called the polariser, the second the analyser.

Fresnel first pointed out that by mechanical stress it is possible to impart to glass a property analogous to that of doubly refracting crystals, making its elasticity (for light) different in different directions. Sir David Brewster showed that a piece of glass heated and then suddenly cooled, i.e. unannealed, possesses similar properties. If it be placed between the polariser and analyser of a polariscope and examined by white light, a gorgeous display of colour is observed. The reason is that the plane-polarised light from the polariser on entering the glass is generally split into two rays vibrating in planes at right angles to one another, and these pass through the glass with different velocities.

CHROMATIC EFFECTS OF A PENTAGON OF UNANNEALED GLASS IN POLARIZED LIGHT.

ISOCHROMATIC LINES IN A TRIANGLE OF UNANNEALED GLASS.

When they reach the analyser only that component of each set of vibrations can pass through which takes place in the plane corresponding to transmission through the analyser. Then it may happen that the two rays of light of one particular colour are so vibrating that the waves interfere and destroy one another, so that this colour is completely absent in the transmitted light while other colours are partially destroyed. The result is that the combination of colours which passes through produces a particular tint, and this will be different according to the obliquity of the ray, the thickness of the glass, and the amount of strain it has experienced, for on the latter depends the difference of velocity of the two rays, and consequently the particular colour destroyed in the case of glass of a given thickness.

Maxwell’s first serious experiments on light appear to have had their origin in his visit to Mr. Wm. Nicol in April 1847. After this visit he constructed a polariscope of cardboard, employing blackened glass mirrors as polariser and analyser. This was supplied with lenses for use when a conical beam of polarised light was required. The lenses were mounted in cardboard frames. A very similar instrument constructed by him shortly afterwards, but of wood instead of cardboard, together with the lenses mounted as before on cards, is still preserved in the Cavendish Laboratory. Maxwell, after returning from Mr. Nicol’s, prepared some samples of unannealed glass by heating pieces of thick plate glass to redness and allowing them to cool rapidly. By means of a camera lucida adapted to the cardboard polariscope he observed and faithfully copied in water-colours some of the chromatic effects exhibited by these plates of glass, showing the manner in which the glass was strained by the rapid cooling. Some of these figures are shown in Plate III., and will be again referred to.[251] As mentioned in the previous part of this work, of these water-colour drawings he sent some to Nicol, who presented him in return with a pair of Nicol prisms of his own construction, which are now, with the original colour-top, polariscope, specimens of unannealed glass, etc., in the Cavendish Laboratory. His experiments on the passage of light through solids exposed to strain suggested to Maxwell the employment of polarised light as an analyser of the strains in the different portions of an elastic solid when exposed to mechanical stress, and led to the production of the paper on “Elastic Solids” read before the Royal Society of Edinburgh in February 1850; but before giving an account of this paper we must mention another investigation connected with the physiology of vision.

There are some persons who, when they look at a point in the sky distant from the sun, at once observe two conspicuous yellow brushes with their axis in a plane passing through the sun, while the space between exhibits the complementary violet colour. This phenomenon was first noticed by Haidinger in 1844, and is known as Haidinger’s Brushes. The appearance is only transitory, and disappears in a very short time if the eye he kept directed to the same point in the heavens. The same appearance is produced whenever “plane-polarised” light enters the eye; as, for example, when light is reflected from a polished surface of glass at a particular angle. But there are some persons, on the other hand, who are apparently incapable of seeing Haidinger’s Brushes, or only see them with difficulty. Generally it is persons of a dark complexion who see Haidinger’s Brushes more readily than others, and in such persons the yellow spot appears to be peculiarly insensitive to blue light. A paper on this subject was read before the mathematical and physical section of the British Association by Professor Maxwell in 1866 (Report, Part II.) He found that on looking through a solution of chrome alum, the centre of the field of view appears distinctly pink and much paler than the rest of the solution on account of this peculiarity of the foramen centrale. The effect decreases if the observer continue to look. Maxwell’s attention was first called to this peculiarity of his own eye by noticing a black hand in the blue portion of the spectrum whenever he directed his eye straight to that portion. The hand never appeared in any other part of the spectrum, but followed the eye up and down the spectrum in the blue, vanishing as soon as the optic axis passed out of the blue into other colours.

The following account of Maxwell’s explanation of Haidinger’s Brushes, and the personal reminiscences which accompany it, have been kindly contributed by Professor William Swan:—

My earliest noteworthy recollection of Clerk Maxwell dates from 1850, when the British Association was in Edinburgh, and its “Section A” used to meet in the natural philosophy class-room of the University. That year communications were made by Sir David Brewster and Professor Stokes on the remarkable phenomenon of vision discovered by Haidinger in 1844, and since known as “Haidinger’s Brushes.” One day when a paper had been read, which, if I remember rightly, was Brewster’s, a young man rose to speak. This was Clerk Maxwell. His utterance then, most likely, would be somewhat spasmodic in character, as it continued to be in later times, his words coming in sudden gushes with notable pauses between; and I can well remember the half-puzzled, half-anxious, and perhaps somewhat incredulous air, with which the president and officers of the section, along with the more conspicuous members who had chosen “the chief seats” facing the general audience, at first gazed on the raw-looking young man who, in broken accents, was addressing them. For a time I was disposed to set down his apparent embarrassment to bashfulness; and such, I daresay, was the general impression. Bashful, very likely, in some degree he was. But, at all events, he manfully stuck to his text; nor did he sit down before he had gained the respectful attention of his hearers, and had succeeded, as it seemed, in saying all he meant to say.

My only tolerably distinct further recollection is that he handed in a small piece of apparatus which he had made by cementing together on glass sectors of sheet gutta-percha, these being so cut out of the sheet and put together that its fibrous structure, due to rolling in the process of manufacture, should radiate, at least approximately, from a central point. This arrangement, as I understood at the time, in polarised light, reproduced, or rather simulated, Haidinger’s phenomenon. But after an interval of thirty years I must speak with caution. Brewster (British Association Report, 1850), ascribed Haidinger’s phenomenon to “polarising structure existing in the cornea and crystalline lens, as well as in the tissues which lie in front of the sensitive layer of the retina;” while Stokes proved that, when the variously coloured rays of the prismatic spectrum were admitted separately into the eye, in the blue rays alone could Haidinger’s Brushes be seen. A few years later, in his paper “On the Unequal Sensibility of the Foramen Centrale to Light of Different Colours” (Brit. Assoc. Report, 1856), Clerk Maxwell says that, on looking through a prism at a long vertical slit he saw an elongated dark spot running up and down the spectrum, but refusing to pass out of the blue into the other colours. This appearance, he concludes, is due to the “Foramen centrale” of Soemmering; and he adds that when a Nicol’s prism is employed the brushes of Haidinger are well seen in connection with the spot. The appearance of a dark spot on a blue ground only, Maxwell then, or at least afterwards, believed to be due to the yellow pigment of the macula lutea, of which the fovea centralis, as its name imports, is the middle portion, and, as is well known, the place of most distinct vision, having a special selective absorption for the blue rays. His notable discovery that Haidinger’s Brushes were only to be seen in connection with the shadow of the yellow spot thus pointed conclusively to the spot itself as the seat of the phenomenon; or, as he himself puts it, makes evident the fact that the brushes are the spot analysed by polarised light. Having thus, to his own satisfaction, localised the polarising structure concerned in the production of Haidinger’s phenomenon, he seems to have rested, for I am not aware that he ever wrote again on the subject. No such writing at least appears in the Royal Society catalogue of scientific memoirs. Possibly he hoped that some day he might himself examine the actual structure of the yellow spot; or he may have waited for the result of such examination by other hands than his own. Helmholtz, who cites Maxwell’s paper of 1856, attributes the phenomenon of the brushes to a radiating fibrous structure, which, it seems, has actually now been ascertained to exist in the fovea centralis, which he assumes to be feebly polarising, and to possess a special selective absorbing power for the blue rays. (Helmholtz,

Plate IV. DIAGRAM SHEWING THE COLOURS EXHIBITED BY A PLATE OF GELATINE WHEN EXPOSED TO A TORSIONAL SHEAR.

Optique Physiologique, 1867, pp. 548-554.) All Maxwell’s conclusions regarding Haidinger’s Brushes, seem thus to be definitely verified.

William Swan, Ardchapel, Helensburgh, 2d April 1882.

2. In the paper read before the Royal Society of Edinburgh on Feb. 18, 1850, Maxwell describes the mathematical results of the application of Stokes’s theory of elasticity to a number of cases of the deformation of elastic solids, which results, when possible, he tested experimentally by subjecting the strained solid to analysis by polarised light. The first, and perhaps the most interesting, example given in this paper, refers “to the case of a hollow cylinder, of which the outer surface is fixed, while the inner surface is made to turn through a small angle.” The conclusions derived we give in Maxwell's own words:—

Therefore, if the solid be viewed by polarised light (transmitted parallel to the axis), the difference of retardation of the oppositely polarised rays at any point in the solid will be inversely proportional to the square of the distance from the axis of the cylinder, and the planes of polarisation of these rays will be inclined to the radius at that point.

The general appearance is, therefore, a system of coloured rings, arranged oppositely to the rings in uniaxial crystals, the tints ascending in the scale as they approach the centre, and the distance between the rings decreasing towards the centre. The whole system is crossed by two dark bands inclined to the plane of primitive polarisation, when the plane of the analysing plate is perpendicular to that of the first polarising plate (see Plate IV.)

A jelly of isinglass poured, when hot, between two concentric cylinders, forms, when cold, a convenient solid for this experiment; and the diameters of the rings may be varied at pleasure by changing the force of torsion applied to the interior cylinder.

By continuing the force of torsion while the jelly is allowed to dry, a hard plate of isinglass is obtained, which still acts in the same way on polarised light, even when the force of torsion is removed.

It seems that this action cannot be accounted for by supposing the interior parts kept in a state of constraint by the exterior parts, as in unannealed and heated glass; for the optical properties of the plate of isinglass are such as would indicate a strain pressing in every part of the plate in the direction of the original strain, so that the strain on one part of the plate cannot be maintained by an opposite strain on another part.

Two other uncrystallised substances have the power of retaining the polarising structure developed by compression. The first is a mixture of wax and resin, pressed into a thin plate between two plates of glass, as described by Sir David Brewster in the Philosophical Transactions for 1815 and 1850.

When a compressed plate of this substance is examined with polarised light, it is observed to have no action on light at a perpendicular incidence; but when inclined it shows the segments of coloured rings. This property does not belong to the plate as a whole, but is possessed by every part of it. It is, therefore, similar to a plate cut from a uniaxial crystal perpendicular to the axis.

I find that its action on light is like that of a positive crystal, while that of a plate of isinglass, similarly treated, would be negative.

The other substance which possesses similar properties is gutta-percha. This substance in its ordinary state, when cold, is not transparent even in thin films; but if a thin film be drawn out gradually, it may be extended to more than double its length. It then possesses a powerful double refraction, which it retains so strongly that it has been used for polarising light. As one of its refractive indices is nearly the same as that of Canada balsam, while the other is very different, the common surface of the gutta-percha and Canada balsam will transmit one set of rays much more readily than the other, so that a film of extended gutta-percha placed between two layers of Canada balsam, acts like a plate of nitre treated in the same way. That these films are in a state of constraint may be proved by heating them slightly when they recover their original dimensions.

Some pieces of gutta-percha mounted in this way by Professor Maxwell are still preserved in the Cavendish Laboratory; as are also the original plates of isinglass above referred to. The interior cylinder employed for twisting these plates was a cork, and in one of the plates two corks were placed with their circumferences about in. apart, and were twisted equally in the same direction. The result of this operation is described as case XIII. in the paper in question, and is determined geometrically by the superposition of the two conditions of strain due to the two twists independently.

Fig. 4.

The isochromatic curves thus obtained are represented in Fig. 4, which is taken from that in the Edinburgh Transactions. The points , , correspond to no retardation. These curves completely agree with those obtained when the plate of isinglass is examined with circularly polarised light. The advantage of employing circularly polarised light lies in the fact that whatever may be the direction of the lines of stress in the isinglass the two component rays into which the light may be supposed separated on entering the medium are of equal intensity, and the colour therefore depends only on the state of strain and not on the position of the plane of primitive polarisation.

The last example of stress in an elastic solid is that of a triangle of unannealed glass, in which “the lines of equal intensity of the action on light are seen without interruption by using circularly polarised light.” They are represented in Plate III. In Figs. 5 and 6 , , , are the neutral points, or points of no action on light, and , , are the points where the action is greatest; and the intensity of the action at any other point is determined by its position with respect to the isochromatic curves.

“The direction of the principal axes of pressure at any point is found by transmitting plane polarised light, and analysing it in the plane perpendicular to that of polarisation. The light is then restored in every part of the triangle, except in those points at which one of the principal axes is parallel to the plane of polarisation. A dark band formed of all these points is seen, which shifts its position as the triangle is turned round in its own plane.

Fig. 5.

Fig. 5 represents these curves for every fifteenth degree of inclination. They correspond to the lines of equal variation of the needle in a magnetic chart.

“From these curves others may be formed which shall indicate, by their own direction, the direction of the principal axes at any point. These curves of direction of compression and dilatation are represented in Fig. 6; the curves whose direction corresponds to that of compression, are concave toward the centre of the triangle, and intersect at right angles the curves of dilatation.”

The figures on Plate III. are copies of water-colour sketches made by Professor Maxwell in the very early periods of these investigations, and show the isochromatic lines in unannealed glass in the case of a pentagon and a triangle.

Fig. 6.

The remainder of the paper on elastic solids is taken up with a discussion of a number of examples of great importance to the engineer, such as the flexure of beams, the torsion of cylinders, and the like. To these engineering problems Maxwell again returned many years later, and his paper “On Reciprocal Figures, Frames, and Diagrams of Force,” read before the Royal Society of Edinburgh, 7th February 1870, received the award of the Keith Medal. The Theory of Oërsted’s Piezometer, given in the paper of 1850, is, however, of great scientific value. In it Maxwell points out that the behaviour of a vessel exposed to equal pressures within and without depends only on its cubic compressibility, while the relation between the cubic compressibility and the rigidity in solids, so far from being constant, as Poisson supposed, may be almost any whatever, cork having great rigidity in comparison with its power to resist compression, while caoutchouc, on the other hand, has extremely little.

3. While a bachelor-scholar of Trinity College, on March 13, 1854, Clerk Maxwell read a paper before the Cambridge Philosophical Society “On the Transformation of Surfaces by Bending.” This paper, which embodies a great deal of thought, and indicates that its author possessed an extensive acquaintance with the geometrical works of Gauss, Monge, Liouville, and others, is of more interest to the pure mathematician than to the general reader, but in connection with it we may mention a surface which was prepared for Professor Maxwell many years afterwards, and coated by his own hands, as described below, and now preserved in the Cavendish Laboratory. If a heavy uniform string be suspended from two points, and allowed to hang freely, it assumes the form of a curve called a catenary. If a board be cut out in this shape, and the string be stretched around its edge, then cut at the lowest point and one-half of the string unwrapped so as always to be kept stretched, and therefore the free portion of the string a tangent to the board at the point where it leaves it, the extremity of the string will describe a curve called a tractory, because it is the curve traced by a particle lying on a rough horizontal plane, when the end of a string attached to it describes a straight line on the plane, the particle starting from a point outside the straight line. As we go farther along the curve traced by the particle, we of course continually approach the straight line described by the end of the string, but never reach it. This straight line is the “asymptote” of the tractory, and is also the “directrix” of the catenary above referred to. If the tractory be made to revolve about its asymptote it will trace out a surface which has the peculiarity that the intrinsic curvature is the same at every point of the surface and is negative, the two principal radii of curvature being opposite in direction and inversely proportional the one to the other. If a flexible surface be constructed so as to fit a portion of the surface of the solid, thus forming a coat, it will be found to fit equally well at every part of the surface, and it may be inverted or turned through any angle whatever, and it will continue to fit every portion of the solid equally well. By weaving together curved strips of parchment paper and pasting them, Professor Maxwell constructed a coat which fitted every portion of the solid, however it might be placed upon it.

Maxwell’s fondness for geometry has already been alluded to more than once, and not unfrequently he would take up geometrical questions of a very generalised character. A simple and very good illustration of some of his investigations in this department is to be found in a paper “On Hills and Dales,” published in the Philosophical Magazine for December 1870. In this paper, having described the general character of contour-lines, or the lines in which the surface of the earth is cut by level surfaces. Maxwell proceeds to show[252] that the number of summits, or regions of elevation reduced to points, exceeds by one the number of passes, a pass being the point where two regions of elevation unite. Similarly, the number of Bottoms, or regions of depression, exceeds by one the number of Bars, a Bar being the boundary between two regions of depression. Lines of slope are defined as lines everywhere perpendicular to the contour lines. By following lines of slope we generally reach a Summit or a Bottom, but we may reach a Pass or Bar. “Districts whose lines of slope run to the same bottom are called Basins or Dales. Those whose lines of slope come from the same summit may be called, for want of a better name, Hills.” “Dales are divided from each other by Watersheds, and Hills by Watercourses.” “Lines of Watershed are the only lines of slope which do not reach a bottom, and lines of Watercourse are the only lines of slope which do not reach a summit.” These extracts indicate Maxwell’s anxiety that definite meanings should be assigned to all scientific terms, and that they should be used only in the senses so defined. A further illustration of this trait in his character is found in his paper “On the Mathematical Classification of Physical Quantities,” published in The Proceedings of the London Mathematical Society, vol. III. No. 34. We give one other example, taken from Matter and Motion, a little work published by the S. P. C. K., and a model of scientific accuracy and philosophic thought. “When the simultaneous values of a quantity for different bodies or places are equal, the quantity is said to be uniformly distributed in space.” “When the successive values of a quantity for successive instants of time are equal, the quantity is said to be constant.” Such examples might be multiplied almost indefinitely from Maxwell’s published writings. The more elementary the work the greater need did he see for exactness in definitions and accuracy in the use of words; and perhaps no one has done more than Maxwell in giving accuracy to the expression of scientific thought. His influence in this respect has especially been felt by all who have attended the Physical School in Cambridge. On one occasion he remarked, in his half-humorous way, that spheres are inclusive figures but circles are exclusive:—people are always trying to get into other circles, but to do so they must get out of their own spheres.

4. The keystone as well as the foundation of physical science is dynamics—a subject to which Maxwell was greatly attracted in early days, and to which, even in its most elementary principles, he constantly reverted in after life. We have already referred to his Dynamical Theory of the Electromagnetic field; the sections of his Treatise on Electricity and Magnetism, which are devoted to dynamical principles and the general equations of motion, form a most valuable compendium of dynamics; while for those who require less strong meat, his treatment of the subject in the Theory of Heat leaves little to be desired; and last, but not least, under the comprehensive title of Matter and Motion, we have a treatise on dynamics written for children in the higher grade schools, so simple that the most casual reader will think that he understands it, while it is the admiration of professed mathematicians, and by no means easy reading for those who expect, or have gained, high places in the mathematical tripos: but as the book is well within the reach of every reader, it is unnecessary here to attempt any detailed account of it.

But Maxwell’s investigations in dynamics were not confined to paper. Of the problems suggested to him by the devil on two sticks we have no account. During his residence in Cambridge he endeavoured to investigate the process by which a cat is enabled invariably to alight on her feet. The mode of conducting the experiments and the impression they left on the mind of the College will appear from the following extract from a letter written to Mrs. Maxwell, from Trinity, on January 3d, 1870, when Professor Maxwell was examining for the Mathematical Tripos:—

There is a tradition in Trinity that when I was here I discovered a method of throwing a cat so as not to light on its feet, and that I used to throw cats out of windows. I had to explain that the proper object of research was to find how quick the cat would turn round, and that the proper method was to let the cat drop on a table or bed from about two inches, and that even then the cat lights on her feet.

The “Dynamical Top,” which wag invented by Maxwell to illustrate dynamical propositions, technically so-called, was, in its final form, constructed of brass by Mr. Ramage of Aberdeen. It was this top which Maxwell brought with him to Cambridge when he came up for his M.A. degree in the summer of 1857, and exhibited to a tea-party in his room in the evening. His friends left it spinning, and next morning Maxwell, noticing one of them coming across the court, leapt out of bed, started the top, and retired between the sheets. It is needless to say that the spinning power of the top commanded as great respect as its power of illustrating Poinsot’s Theorie Nouvelle de la Rotation des Corps.

In the work just referred to, Poinsot shows that a body supported at its centre of gravity and rotating freely about it will move in the same manner as an ellipsoid whose centre is fixed, but which rolls in such a way as always to rest against a fixed plane (the invariable plane), which is, of course, a tangent plane at the point of contact. The line from the centre to the point of contact is the axis about which the body is at the instant rotating, and this line will, unless the plane touch the ellipsoid at the extremity of one of the principal axes, move both in space and in the body. The curve which the extremity of the axis of rotation describes on the invariable plane is called a herpolhode, while that which it describes on the surface of the ellipsoid is called a polhode. If at any instant the axis of rotation be very near either to the greatest or least axis of the ellipsoid, it will always remain very near that axis, and the polhode will be a small closed curve and the rotation will be stable; but if the axis of rotation be near to the mean axis of the ellipsoid, the polhode will be a very large curve, the axis of rotation will in time deviate very widely from its original position in the body, and the motion will be unstable.

Maxwell’s top consisted of a brass bell, a long screw passing through the top of the bell and terminating in a steel point “finished without emery and afterwards hardened,” serving as the axle. The point rested in an agate cup on the top of a pillar. A heavy nut could be screwed up and down on the axle for coarse adjustments, and the axle could be screwed through the top of the bell, to bring the point to coincide with the centre of gravity of the mass. The moments of inertia could be altered by means of the nut above referred to, and by nine screws with massive milled heads, six of which screwed horizontally into the rim of the bell, while three were arranged symmetrically around the top of the rim, and admitted of a vertical motion. By means of these screws the axle could be made the axis of greatest, least, or mean moment of inertia, or not a principal axis at all. The axle terminated in a point, by which a coloured disc could be fixed upon it. “The best arrangement, for general observations, is to have the disc of card divided into four quadrants, coloured with vermilion, chrome yellow, emerald green, and ultramarine. These are bright colours, and if the vermilion is good they combine into a grayish tint when the revolution is about the axle, and burst into brilliant colours when the axis is disturbed. It is useful to have some concentric circles, drawn with ink, over the colours, and about twelve radii drawn in strong pencil lines. It is easy to distinguish the ink from the pencil lines, as they cross the invariable axis, by their want of lustre. In this way the path of the invariable axis may be identified with great accuracy, and compared with theory.” If the top revolve about its axle, the whole disc will appear gray, but when the axle moves, the quadrants in which the “invariable axis” lies will be indicated by a circle of the corresponding colour appearing in full purity. Whenever the invariable line crosses an ink or pencil line, a dark ink dot or lustrous pencil dot will appear at the centre of the circle, and when it passes into another quadrant the circle will contract, then change its colour, and expand again. The use of the nine screws is thus indicated by Professor Maxwell:—“There must be three adjustments to regulate the position of the centre of gravity, three for the magnitude of the moments of inertia, and three for the directions of the principal axes—nine independent adjustments, which may be distributed as we please among the screws of the instrument.” A full account of the top and the mode of using it, together with a note on the rotation of the earth, will be found in the Transactions of the Royal Society of Edinburgh, vol. XXI., from which the above quotations have been taken.

5. In 1610 Galileo turning his telescope to Saturn, observed a projection on each side of the planet, which led him to conclude that “Saturn consists of three stars in contact with one another.” In 1659 Huygens discovered the true nature of these appendages, which, by their continually varying forms, had puzzled previous astronomers, and concluded that Saturn is girded with a thin flat ring inclined to the ecliptic; its external diameter being about times that of the planet itself. In 1665 William Bell observed a dark line running round the northern surface of the ring. In 1675 Cassini noticed the same on the other surface, and concluded that the ring consists of two concentric rings, of which the inner is the brighter. Hadley observed the shadow of the planet thrown upon the surface of the ring, and the shadow of the ring on the planet, and showed that Saturn rotates in the same plane with the rings. In 1714 Moradi observed a want of symmetry in the ring on opposite sides of the planet, and found that when it had disappeared on the eastern side it was still visible on the western, but he did not discover the cause of the phenomena. The observations of Sir William Herschel, published in 1790, corroborated the opinion of Cassini respecting the division. By observing certain spots on the surface of the planet, Herschel found that it rotated on its axis in 10 hours 29 minutes 16·8 seconds in the same direction as the rotation of the earth. In 1789, as the earth passed through the plane of Saturn’s ring, Herschel noticed some bright spots on the edge of the ring which were carried almost to the end of the diameter, and appeared to rotate about Saturn in about 10 hours minutes.

The observations of the present century show that the outer of the two bright rings is permanently divided into two concentric rings by a very narrow gap, and when their plane is inclined to the line of sight at a considerable angle, so that the rings are widely open, a series of dark elliptic curves near the extremities of the major axis of the elliptic projection indicates that each ring is still further broken up into a number of thin concentric rings, the gaps between them being apparently filled by the edges of the inner rings except near the extremity of the major axis of the projections.

In 1850 a dark ring encircling Saturn within the other two rings was discovered independently by three astronomers in England and America. The distance between the outer edge of the dark ring and the inner edge of the adjoining bright ring appeared to vary between narrow limits, the rings sometimes appearing to be in contact. This innermost ring is transparent, so that the edge of the planet can be seen through it, and as it is seen in all positions without distortion it seems that the ring cannot consist of a transparent gas or liquid. The rings appear to be much thicker near the planet than at their outer edges. A comparison of measurements of the diameters of the rings at different times seems to indicate a rapid change in their breadths, the circumference of the outer ring extending outwards, while the inner bright ring appears to be approaching the planet, so that the whole breadth of the ring system is increasing. At least this is the conclusion to which Struve arrived from a comparison of his measurements with those of Huygens and Herschel.[253] According to Hind the exterior diameter of the outer ring is about 170,000 miles, its interior diameter 150,000; the exterior diameter of the inner bright ring is 147,000 miles, its interior diameter 114,000 miles; and the equatorial diameter of the planet 75,500 miles; but, as above stated, the breadth of the rings appears to be increasing. Sir John Herschel was of opinion that the thickness of the rings did not exceed 100 miles, and Bessel calculated that their mass, as determined by the disturbance they produced in one of Saturn’s satellites, did not exceed th the mass of the planet.

If Saturn’s rings were solid and at rest, but of course subject to the attraction of the planet, the stress upon them would be such that we can conceive of no material capable of sustaining it. Maxwell remarked that iron would not only be plastic but semi-fluid under such stresses. By allowing the rings to rotate, the attraction of the planet might expend itself in producing the necessary acceleration towards the centre, or, as is generally stated, might be balanced by the centrifugal force, and the stress might be thus relieved to a great extent; but if the rings rotated with the velocity proper to the interior of the inner ring, the rest would tend to fly off into space, while if they rotated with the velocity proper to the exterior of the outer ring, the inner portions would tend to fall towards the planet; and even if they rotated with the velocity proper to any intermediate portion of the system, the parts beyond would tend to fly outwards, and the parts within would tend to fall towards the planet, unless this tendency were compensated by the attraction of the ring itself. But there is another objection to the hypothesis of the rings being uniform solids, viz. the instability of the motion. If the motion of the ring were slightly disturbed, as is always the case, it would not return to its original orbit, but its deviation would increase until it came in contact with the planet itself.

Laplace was the first to investigate the conditions of stability of Saturn’s ring system. He concluded that if the rings are solid, they must consist of a great number of very small concentric rings, each rotating independently with its proper velocity about the planet. The velocity calculated by Laplace for the circumference of the outer ring agreed with that determined, as above mentioned, by Herschel. Laplace avoided the difficulty of the instability of the rings above alluded to by supposing that they were not homogeneous, but that their centres of gravity were at some distance from their geometrical centres. Laplace also showed that, for the attraction of the ring to neutralise its tendency to split up, it is necessary that its density should be that of the planet. In 1851 Professor Pierce showed that the number of rings in the system must be very much greater than Laplace had supposed. This was the position of the question when it was taken up by Maxwell.

On March 23d, 1855, the Examiners announced the subject for the Adams Prize in the following terms:—

The University having accepted a fund raised by several members of St. John’s College, for the purpose of founding a prize to be called the Adams Prize, for the best essay on some subject of pure mathematics, astronomy, or other branch of natural philosophy, the prize to be given once in two years, and to be open to the competition of all persons who have at any time been admitted to a degree in this University:—

The Examiners give notice, that the following is the subject for the prize to be adjudged in 1857:—

The Motions of Saturn's Rings.

The Problem may be treated on the supposition that the system of Rings is exactly, or very approximately, concentric with Saturn, and symmetrically disposed about the plane of his equator, and different hypotheses may be made respecting the physical constitution of the Rings. It may be supposed (1) that they are rigid; (2) that they are fluid, or in part aeriform; (3) that they consist of masses of matter not mutually coherent. The question will be considered to be answered by ascertaining, on these hypotheses severally, whether the conditions of mechanical stability are satisfied by the mutual attractions and motions of the Planet and the Rings.

It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearances both of the bright rings and the recently discovered dark ring may be most satisfactorily explained; and to indicate any causes to which a change of form, such as is supposed from a comparison of modern with the earlier observations to have taken place, may be attributed.

In his essay, to which the prize was adjudged, and which occupies sixty-eight pages of quarto, Maxwell considered in the first place the hypothesis of Laplace, and showed that to secure stability the irregularity of the ring must be enormously great. Speaking of Laplace’s conclusion that the rings must be irregular, Maxwell says:—

We may draw the conclusion more formally as follows:— If the rings were solid and uniform, their motion would be unstable, and they would be destroyed; but they are not destroyed, and their motion is stable, therefore they are either not uniform or not solid.

I have not discovered, either in the works of Laplace or in those of more recent mathematicians, any investigation of the motion of a ring either not uniform or not solid. So that, in the present state of mechanical science, we do not know whether an irregular solid ring, of a fluid or disconnected ring, can revolve permanently about a central body; and the Saturnian system still remains an unregarded witness in heaven to some necessary but as yet unknown development of the laws of the universe.

The following extract shows how Maxwell mentally realised every problem which he considered:—

When we have actually seen that great arch swung over the equator of the planet without any visible connection, we cannot bring our minds to rest. We cannot simply admit that such is the case, and describe it as one of the observed facts in nature, not admitting or requiring explanation. We must either explain its motion on the principles of mechanics, or admit that, in the Saturnian realms, there can be motion regulated by laws which we are unable to explain.

The investigation of the stability of the motion of a solid ring, otherwise uniform but loaded with a heavy particle upon its circumference, led to the conclusion that the mass of the particle must be about four and a half times that of the rest of the ring—

But this load, besides being inconsistent with the observed appearance of the rings, must be far too artificially adjusted to agree with the natural arrangements observed elsewhere, for a very small error in excess or defect would render the ring again unstable.

We are therefore constrained to abandon the theory of the solid ring, and to consider the case of a ring the parts of which are not rigidly connected, as in the case of a ring of independent satellites, or a fluid ring.

There is now no danger of the whole ring, or any part of it, being precipitated on the body of the planet. Every particle of the ring is now to be regarded as an independent satellite of Saturn, disturbed by the attraction of a ring of satellites at the same mean distance from the planet, each of which, however, is subject to slight displacements. The mutual action of the parts of the ring will be so small compared with the attraction of the planet that no part of the ring can ever cease to move round Saturn as a satellite.

But the question now before us is altogether different from that relating to the solid ring. We have now to take account of variations in the form and arrangement of the parts of the ring, as well as its motion as a whole, and we have as yet no security that these motions may not accumulate till the ring entirely loses its original form, and collapses into one or more satellites circulating round Saturn. In fact such a result is one of the leading doctrines of the “nebular theory” of the formation of planetary systems; and we are familiar with the actual breaking up of fluid rings under the action of “capillary force,” in the beautiful experiments of M. Plateau.

In this essay I have shown that such a destructive tendency actually exists, but that by the revolution of the ring it is converted into the condition of dynamical stability.

The investigation of the motion of a ring composed of a continuous liquid, led to the result that the waves set up in the ring would cause it to break up into a number of drops which might revolve as satellites about Saturn; but if the condition of stability was not fulfilled, they would coalesce, forming a smaller number of larger drops, until the condition of stability was secured; and the same would be the case if the ring were supposed to consist of a number of concentric narrow rings. The liquid ring, therefore, did not afford a more satisfactory result than the solid one, though it appeared that the internal friction would produce no sensible effect, and Maxwell had recourse to “the dusky-ring, which is something like the state of the air supposing the siege of Sebastopol conducted from a forest of guns 100 miles one way and 30,000 miles the other, and the shot never to stop but go spinning away round a circle radius 170,000 miles.”[254] This, however, is not the order in which the investigations appear in the paper before us.

Having dismissed the assumption of the solid rings, Maxwell investigated the effect of disturbance on a ring of small transverse section, the parts of which are not rigidly connected. The disturbances he treated as made up of “harmonic” elements, according to Fourier’s method; and, first treating the ring as a number of equal satellites revolving round the planet, he found that such a ring “can always be rendered stable by increasing the mass of the central body and the angular velocity of the ring.” If the ring consist of 100 satellites, the mass of the planet must exceed 4352 times that of the ring in order to secure stability for all displacements. “If this condition be not fulfilled,... then, although the motion depending upon long undulations may remain stable, the short undulations will increase in amplitude till some of the neighbouring satellites are brought into collision.” The satellites of such a ring, when the condition of stability is fulfilled, admit of four vibrations in different periods in ellipses about their mean positions, and these vibrations will be transmitted as waves with different velocities round the ring, so that the form of the ring at any instant resembles “that of a string of beads forming a re-entering curve, nearly circular, but with a small variation of distance from the centre,” forming a number of “regular curves of transverse displacement at regular intervals round the circle. Besides these, there are waves of condensation and rarefaction, the effect of longitudinal displacement.” Any external disturbance, such as a satellite or wave in another ring, will produce a forced wave in the ring, and if the angular velocity of the external disturbing cause around the ring coincide, or nearly coincide, with the velocity of one of the free waves of the ring, the amplitude of the forced wave will increase indefinitely until the ring is destroyed; but if this condition is not fulfilled, the forced wave will accompany the disturbing cause around the ring as the tide follows the moon.

The effect of one ring upon another is to produce in it a series of forced waves travelling with the same angular velocity as the free waves in the disturbing ring. Hence in a system of two rings “there will be eight waves in each ring, and the corresponding waves in the two rings will act and react on each other, so that, strictly speaking, every one of the waves will be in some measure a forced wave, although the system of eight waves will be the free motion of the two rings taken together.”

The dynamical stability of a ring of satellites is explained by the consideration that when one of the satellites, in consequence of its oscillations, moves with more than its mean velocity, and thus gets in front of its proper position, it will be carried farther away from the planet, and thus not only will its linear velocity be diminished, but, in virtue of its increased distance, its angular velocity will be still farther diminished and, lagging behind, it will fall back into its proper position and, oscillating through it with a velocity less than its mean, will approach the planet and the reverse action will take place.

After considering the manner in which the ring will break up if the condition of stability is not fulfilled, Maxwell takes up the question of “the dusky ring,” consisting of innumerable small particles resembling “a shower of rain, hail, or cinders.” For the stability of such a ring its average density must not exceed of that of the planet, and the density of Saturn being only ·7505, it follows that the average density of the ring cannot greatly exceed that of air at ordinary pressure. Laplace showed that for a ring to rotate as a whole with uniform velocity about Saturn, the density of the planet cannot exceed 1·3 times that of the ring. Hence the particles must move independently, or in a series of concentric rings.

In the case of concentric rings of satellites, their mutual disturbances will destroy one another if the velocity of any one of the four free waves of one coincide, or nearly coincide, with that of any free wave in another; and it is impossible that there should be any great number of rings without this condition frequently recurring.

In the case of a large number of concentric rings, the stability of each pair must be investigated separately, and if in the case of any two, whether concentric rings or not, there are a pair of conspiring waves, those two rings will be agitated more and more till waves of that kind are rendered impossible by the breaking up of those rings into some different arrangement. The presence of the other rings cannot prevent the mutual destruction of any pair which bear such relations to each other.

It appears, therefore, that in a system of many concentric rings there will be continually new cases of mutual interference between different pairs of rings. The forces which excite these disturbances being very small, they will be slow of growth, and it is possible that by the irregularities of each of the rings the waves may be so broken and confused as to be incapable of mounting up to the height at which they would begin to destroy the arrangement of the ring. In this way it may be conceived to be possible that the gradual disarrangement of the system may be retarded or indefinitely postponed.