On the Action of Crystallized Bodies on Homogeneous Light, and on the Causes of the Deviation from Newton's Scale in the Tints Which Many of Them Develope on Exposure to a Polarised Ray

IV. On the action of crystallized bodies on homogeneous light, and on the causes of the deviation from Newton's scale in the tints which many of them develope on exposure to a polarised ray. By J. F. W. Herschel, Esq. F. R. S. Lond. and Edin. Read December 23, 1819. Since the period of the brilliant discovery of Malus of the polarisation of light by reflection, the investigation of the general laws which regulate the action of crystallized bodies on light, has advanced with a rapidity truly astonishing, and the labours of an Arago, a Brewster, and a Biot, have already gone far towards completing the edifice of which that distinguished philosopher laid the foundation. When Malus wrote, the list of doubly refracting crystals was small, and the most remarkable among them possessing only one axis of double refraction, it seems to have been for some time, tacitly at least, presumed that the law discovered by Huygens, and since re-established in the most rigorous manner for that one,* might hold good in all. The discovery, by Dr. Brewster, of crystals possessing two axes of double refraction, or two * The author of the article on Polarisation, in the 63d Number of the Edinburgh Review, just published, is guilty of a most unpardonable mistake, in asserting, (p. 188), as deducible from Dr. Brewster's experiments, that the Huygenian law is incorrect, for carbonate of lime. Dr. Brewster's general formulae for crystals with two axes resolve themselves into the Huygenian law when the axes coincide, of which case it is only an extension. That excellent philosopher, if I understand English, in the paragraph which gave rise to this strange assertion, only means to declare his opinion that it remains undemonstrated. directions in which a ray may penetrate their substance without separation into distinct pencils, has proved the fallacy of any such generalization, and rendered it necessary to enter on a far more extensive scale of investigation. There are two methods which may be pursued in observations on double refraction and polarisation, the one direct, the other indirect. The former turns on immediate observations of the angular deviation of the extraordinary pencil, and is, of course, only applicable when the forces which act exclusively on the rays composing it are sufficiently intense to cause a sensible separation of the two pencils. There exist, however, a multitude of crystals in which the force of double refraction is so feeble as to produce scarcely any, or at most a very inconsiderable deviation of the extraordinary ray, and in which, consequently, the laws of double refraction could neither be investigated nor verified, without having recourse to some artificial means of magnifying the quantity to be observed; a thing easy enough in theory, but requiring, in practice, the greatest nicety on the part of the observer, and in many cases altogether impracticable, from the physical constitution of the crystals themselves. The indirect method depends on the discovery of Arago, scarcely inferior in intrinsic importance to that of Malus, of the separation of a polarised ray into complementary portions by the action of a crystallized lamina. It was reserved, however, for the genius of M. Biot, to trace this striking phenomenon to its ultimate causes, in the action of crystals on the differently coloured rays, and to develope, in a simple and elegant theory, the successive gradations by which the polarisation of a ray in its passage through a doubly refracting crystal is performed; while, on the other hand, the splendid phenomena of the polarised rings, which we owe to Dr. Brewster, have established the connection of the tints so polarised with the force producing the deviation of the extraordinary pencil, and shown the legitimacy of conclusions respecting the intensity of the latter, drawn from observations on the former. This indirect mode of observation, which consists in noticing the gradations of colour for different positions and thicknesses of the crystal, possesses three capital advantages. The first is its extreme sensibility, which enables us to detect the existence, and measure precisely the intensity of forces, far too feeble to produce any measurable deviation of the extraordinary pencil. It, in fact, affords the rare combination of an almost indefinite enlargement of our scale of measurement, with a possibility of applying it precisely to the object measured, arising from the distinctness of all its parts. Another, no less precious, is the leading us by mere ocular inspection to the laws of very complicated phenomena, and enabling us to form, and mould as it were our analytical formulae, not on a laborious, and sometimes deceptive discussion of tabulated measures, but on the actual form of the curves themselves, which are loci of the functions under consideration. It is true, that a reference to tabulated measures is indispensable to give precision to such first approximations; but the power this mode of observation affords of copying our outline fresh from nature, and from the general impression of the phenomena, brought at once under our view, is an advantage not to be despised. Nor ought we, lastly, to omit, in our estimate of advantages, the means thus afforded us of subjecting the minutest fragments of a crystal to a scrutiny as severe as the most splendid specimen, and thus extending our researches to an infinitely greater variety of natural bodies, than we could otherwise hope to examine. In order, however, to render observations on the tints developed by polarised light available, they must be comparable to each other; and it therefore becomes an object of the first importance, to ascertain the existence, and discover the laws of any causes which may operate to disturb their regularity. Ever since I first engaged in experimental enquiries on the polarisation of light, I was struck by the very considerable deviation from the succession of colours in thin laminae, as observed by Newton, which many crystals exhibit when cut into plates perpendicular to one of their axes. I at first attributed this to a want of perfect regularity in their structure, or to inequalities in their thickness, arising from my own inexpertness in grinding and polishing their surfaces; and it was not till habit had rendered me familiar with all the usual causes of deception, that, finding the same phenomena uniformly repeated in different and perfect specimens, my curiosity became excited to enquire into their cause, the more so, as they now began to assume the form of a radical and unanswerable objection to the theory of M. Biot, above alluded to, which affords so perfect an explanation of the tints in crystals with one axis. These phenomena have not escaped the vigilance of Dr. Brewster. In his paper of 1818, he distinctly notices the fact of a deviation from Newton's scale, in crystals with two axes, and promises a more detailed account of it, which however has not yet appeared. But the object of the present communication is not thereby anticipated, as in the only passage in that paper in which he expresses himself otherwise than obscurely on its cause, he appears to regard the deviated tints as analogous to those developed along the axis of rock crystal and by certain liquids; an analogy which, in the present state of our knowledge on that perplexing subject, it seems not easy to admit. In a paper too, which has lately appeared, containing the interesting observations of the same excellent philosopher on the optical structure of the apophyllite, he remarks the very striking deviation of the colours of this crystal from Newton's scale "in the first orders" of its rings; and while he remarks that such deviations are common enough, and indeed universal in crystals "in which the rings are formed by the joint action of two axes," seems to think this analogy close enough to authorize the substitution of two rectangular axes of a negative character for the single positive axis actually observed, according to his own peculiar and ingenious views on this subject. I lost no time in endeavouring to procure a specimen of this mineral, and by the kindness of my friend, Sir Samuel Young, (to whom I owe more than one obligation of this nature) was favoured with one sufficiently transparent for optical examination. From my observations on this body, I think I shall be able to demonstrate satisfactorily, that the phenomena of the apophyllite depend on a principle distinct from that which produces the chief part of the deviation of tints in most crystals with two axes. The course I propose to pursue is, first, to describe the phenomena themselves. I shall then show how these phenomena, complicated as they are in appearance, are all reducible to one very simple and general fact; viz. that the axes of double refraction differ in their position in the same crystal for the differently coloured rays of the spectrum, being dispersed in one plane over an angle more or less considerable, according to the nature of the substance. In many bodies, the magnitude of this dispersion of the axes is comparatively trifling, while in some, not otherwise remarkable for a high ordinary or extraordinary dispersive power, it is enormous, and must render all computation of the tints in which it is not taken into consideration, completely erroneous; and indeed obliterating almost every trace of the Newtonian scale of colour. We have here, then, a new element, which for the future must enter into all formulae of double refraction pretending to rigour, and at the same time are presented with another very striking instance of the inherent distinction between the differently coloured molecules of light, which, since the time of Newton, every new step in optical science has tended to place in a stronger point of view. At the same time, by the easy and complete explanation this principle affords of all the more perplexing anomalies in the tints, the theory of alternate polarisation to which they were hitherto so palpable and formidable an objection, stands relieved from every difficulty, and may now be received as fully adequate to the representation of all the phenomena of the polarised rings, and entitled to rank with the fits of easy transmission and reflection, as a general and simple physical law. In fact, if we investigate by this theory a general analytical expression of the tint developed for any position and thickness of the plate, taking this element into consideration, it will be found to include all the phenomena, as far as they can be computed, while the law of dispersion remains unknown. But we may go yet farther. The nature of the formula furnishes an equation by which the actual quantity of the separation of the extreme red and violet axes may be deduced from observations of the tints of a very simple and accurate nature, being perfectly analogous in principle to the "method of coincidences," which has of late been applied with such success to the most delicate investigations in every department of physical science. The comparison of the results afforded by that equation, with those deduced by direct observation on homogeneous light, while it leaves nothing to desire in point of accuracy, leads to another important result, viz. that the proportionality of the minimum lengths of the periods performed by differently coloured molecules, in a doubly refracting crystal to the lengths of their fits of easy reflection and transmission, supported as it is by an induction of no ordinary extent and accuracy, is yet not universal, admitting a deviation to a very large amount. Hence must of course arise a kind of secondary deviation in the scale of tints. In crystals with two axes, however, this is masked by the much more powerful effect of the separation of the coloured axes; yet even there, is not altogether insensible in an extreme case. In the apophyllite, however, the agency of this secondary cause is placed in the fullest evidence. The application of our general formula to the anomalous tints of that body, while it proves incontestably the exact coincidence of the axes for all the coloured rays, points out at the same time a peculiarity in its action on the more refrangible extremity of the spectrum, of a nature so singular, so entirely without example in all the multitude of natural and artificial bodies hitherto examined, as to render me extremely desirous of prosecuting the research, with the aid of more perfect specimens, and improved methods of observation. Having arrived at the general result of a dispersion of the axes by the sole consideration of the gradation of tints in plates of various thicknesses, it becomes interesting to verify it by direct and independent observation. This I have accordingly done; and the fortunate discovery of a substance in which it is of enormous magnitude, puts it in our power to render the fact sensible to the eye of the most unpractised observer, by an exceedingly simple experiment, to be described in its place. II. Of the general phenomena of crystals which develope tints deviating from Newton's scale, by exposure to polarised light. In describing the phenomena, I shall at present confine myself to the tints developed along the principal section of the crystal, which is supposed placed in an azimuth $45^\circ$ with the plane of primitive polarisation. The observations of the tints in this position are most easily made, and least liable to error, and we shall see presently that it would be superfluous as well as embarrassing to examine other situations, the law of the phenomena being completely deducible from this. In this series of observations, then, we traverse the polarised rings (Pl. V. fig. 1.) in the direction of their axis of symmetry AA', passing through their poles P, P' and centre O. Now if we subject to this examination any one of the following substances: Sulphate of soda? Arragonite, Sulphate of baryta, Sugar, Nitrate of potash, Hyposulphite of strontia, it will be seen that the tints between the poles PP' correspond to lower orders of colour, than would result from assuming P, P', as the origin of the scale, and agree much better with the assumption of certain points \( p, p' \) without the poles, as their zero, or commencement of the scale. The poles themselves, too, instead of being absolutely black, are tinged with colour; and the tints beyond them, instead of descending in the scale from the poles outwards, continue to rise till they reach their maximum (which is a white, more or less brilliant, or an absolute black) at the points \( p, p' \); after which they descend again to infinity. Not that in any case they coincide precisely with the scale of Newton, even with this correction, but, except in extreme cases, approximate to it within some moderate limit of error. If, on the other hand, we examine in the same manner one of the following bodies: Tartrate of soda and potash, Sulphate of magnesia, Borax, Topaz, Mica, it will be found that the imaginary points \( p, p' \) (which we shall call the virtual poles) from which the tints must be reckoned inwards and outwards, to produce the nearest possible agreement with Newton's scale, lie between the poles P, P'. In all these crystals, as the thickness of the plate examined increases, the virtual poles \( pp' \) recede from the actual ones PP', at least in respect of the number of alternations of colour which intervene between them: in other words, the tint deve- loped in the poles, or along the apparent axes of the crystal, descends in the scale of colour, as the thickness of the plate increases, and vice versa. In very small thicknesses, the tints approximate pretty closely to Newton's scale, or wholly coincide with it, while in very great ones, the tint developed in the poles is the composite white of the extremity of the scale. The angular distance, however, of the virtual poles from each other and from the axes, remains absolutely unchanged for all thicknesses; and this striking fact, which I have proved by numerous and satisfactory experiments, was first suggested for examination as a result of theory, and would equally hold good, as will presently be proved, for every conceivable law of double refraction. The substances which I have examined most attentively, are sulphate of baryta, nitre, mica, and Rochelle salt, and the subjoined tables of tints developed for different inclinations in plates of the first and last of these, may serve as examples of the mode of action of the respective classes to which they belong on light, and will afford data for some calculations to follow. The first two columns contain the inclinations corresponding to similar tints of the incident ray on the moveable plate which carries the crystal, in the general apparatus imagined by M. Biot, for observations of this kind. Were the plate cut in a direction precisely perpendicular to the optic axis, (or line bisecting the angle between those of double refraction) and adjusted with perfect accuracy on the instrument, the excesses or defects of these angles above or below $90^\circ$ would represent the angles of incidence. Neither of these conditions were, of course, exactly fulfilled. But it is obvious that the small errors in these particulars (which were ascertained not to exceed $1^\circ$ or $2^\circ$), must affect the computed angles of refraction on both sides of the perpendicular with equal and opposite errors. The same may be said of any error arising from a slight prismaticity of the plate, which, however, must have been extremely small, the plate having always been rendered parallel by the delicate test of the sphærometer, within a very few divisions.* Consequently, in calculating on these data, the mean angle of refraction determined by the simultaneous use of both observations, (their semi-difference being taken for the angle of incidence) may be expected to differ from the truth by an extremely minute quantity. The third column contains the tint developed in the ordinary pencil, and the fourth in the extraordinary. The last notices the remarkable points in the system of rings to which the tints and angles in the other columns correspond. The positions of the poles were determined by interposing a red glass between the crystal examined and the reflector used to polarise the incident light. The glass used for this purpose, was of that kind occasionally found in old church windows, and whose manufacture seems to be numbered among arts now forgotten. It transmits almost the whole of the red rays, and part of the orange, while it completely stops all the more refrangible colours. I have endeavoured in vain to procure a specimen, whose limits of transmission are more confined. Such are said to exist, though very rare, and in the absence of such, the indications of that employed may be taken to correspond to the mean red rays. * Each equal to the $\frac{1}{23809}$th part of an inch. ### Table I. Sulphate of baryta. Thickness of plate = 0.11964. | Corresponding inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |---------------------------|-----------------|--------------|-------------| | | | | | | 43 55 134 37 | Pink | Very pale blue green | Tints beyond the poles. | | 45 133 42 | Bluish green | Pink | | | 46 132 45 | Rich pink | Beautiful green | | | 132 | Whitish | Dull purple | | | 46 58 131 37 | Splendid green | Rich crimson | | | 131 | Blue | Yellow | | | 47 53 130 45 | Crimson | Fine green | | | | Yellow inclining to orange | Blue | | | | Yellow | Purple | | | 48 51 129 30 | Blue green | Rich crimson | | | | Blue | Orange | | | 128 58 | Purple | Pale yellow | | | | Rich crimson red | Pale green or greenish white | | | | Fine orange | Light blue | | | | Pale orange yellow | Dark blue | | | 50 58 127 45 | White | Sombre & very narrow purple | | | | Bluish white | Scarlet or fiery red | | | | Light blue | Orange | | | | Sombre greenish blue | Ruddy white, or very pale | | | 52 0 126 50 | Dirty and very sombre purple | White [orange] | | | | Sombre and narrow pinkish red | White | | | | Very pale violet or pinkish white | Greenish white | | | | White | Dirty bluish green | | | 53 1125 47 | White | Narrow and very sombre violet | Virtual poles | | | White slightly yellowish | Violet [purple]pp', or points of coincidence. | | 125 30 | Pale and dirty olive green | Violet white | | | | Very narrow violet | White | | | 54 8 124 30 | Blue very sombre and narrow | White slightly yellowish | | | | Light blue | Pale yellow | | | | Bluish or greenish white | Indifferent purplish pink | | | 55 25 123 30 | Yellowish white | Sombre and narrow purple | | | | Light yellow | Dark blue | | | | Dull orange pink | Pale greenish blue | | | 56 33 122 15 | Sombre purple | Pale yellow green | | | | Blue | Yellow | | | | Pale green | Fine pink verging to crimson | | | 57 30 | Pale yellow | Purple | | | 57 50 121 0 | Light yellowish pink | Greenish blue | The polesP,P' | | | Rich pink | Bluish green | Tints between the axes. | | 59 12 119 35 | Pale purple | Greenish white | | | | Blue green | Crimson | | | 60 45 118 20 | White | Very pale purple | | | | Pink | Blue green | | | 62 0 117 5 | Pale purple | Whitish | | | 62 58 116 25 | Greenish blue | Pink | | | 63 25 115 35 | White | Dull purple | | | 64 28 114 47 | Pink | Bluish green | | | 65 40 113 2c | Greenish blue | Pink | | | 67 30 111 30 | Pink | Pale greenish blue | | | 69 15 110 0 | Pale greenish blue | Pale pink | | In this plate the virtual poles correspond very nearly to the second minimum of the extraordinary pencil beyond the poles PP'. The same plate was now reduced by grinding to the thickness 0.08816 inches. In this operation, care was taken to grind away the side of the plate most distant from the eye only, leaving the other perfectly untouched and unimpaired in its polish. The plate being reduced to exact parallelism by the sphærometer, was again examined, the same side still remaining next the eye. By this arrangement the same angles of emergence from the posterior surface correspond rigorously to the same directions of the ray in the interior of the crystal, with respect to the axes of its molecules; and thus we avoid completely any errors which might arise from using plates cut at different angles, it being almost impossible to cut two plates precisely alike in this respect. **Table II. Sulphate of baryta. Thickness = 0.08816.** | Corresponding inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |---------------------------|-----------------|---------------|--------------| | 58° 0' 120" | Pink, verging to orange yellow | Blue somewhat greenish | Poles for the mean red rays. | | | Yellow | Dark blue | | | 57° 30' 120" | Pale yellow | Purple | Tints beyond the poles. | | | Greenish white | Pink | | | | Light blue | Yellow verging to orange | | | | Dark blue | Bright yellow | | | 55° 57' 122" | Sombre purple | Yellow white | | | | Very indifferent sombre pink | Bluish white | | | | Pale yellow | Dark indigo blue | | | 54° 5' 124" | White | Sombre violet | | | | Very pale violet white | Dusky greenish yellow | | | 52° 57' 125" | Very sombre violet, almost black | Pure brilliant white | Virtual poles. | | | Sombre and dirty olive green | White rather ruddy | | | | Very pale blue | Orange white | | | 51° 30' 127" | White | Sombre orange or brick red | | | | Orange white | Narrow purple | | | | Bright orange | Blue | | | | Bright scarlet | Pale blue | | | | Narrow crimson | Bluish white | | | 50° 0' 128" | Purple | White | | | | Blue | Yellow | | | | Bluish white | Rich crimson | | | 48° 40' 130" | Yellowish or greenish white | Purple | | | | Orange yellow | Bright blue | | | 47° 35' 131" | Rich crimson | Green | | | | Purple | Good yellow | | | | Bright blue | Pink yellow | | | 46° 7' 132" | Good green | Rich pink or crimson | | | | Rich pink | Splendid green | | | 44° 33' 134" | Greenish purple | Pinkish white | | | | Good green, but pale | Fine pink | | | 43° 35' 135" | Pink red | Greenish blue | | | | Pale bluish green | Pale pink | | | 40° 55' 138" | Pale pink | Very pale greenish blue | | MDCCCXX. In this plate the virtual poles correspond to the second maximum of the extraordinary pencil. It is needless to detail the tints between the poles. The same plate once more reduced, with the same precaution to leave the posterior surface untouched, developed the following series of colours beyond the poles. **Table III. Sulphate of baryta. Thickness = 0.05758.** | Corresponding inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |---------------------------|----------------------------------|--------------------------------|-----------------------------------| | 60° 57' | Fine yellow | Indigo | Poles, for the mean red rays. | | 60° 20' | Pale yellow | Purple | Fints beyond the poles. | | | White inclining to yellow | Dull crimson red | | | | Bluish white | Dull orange | | | | Indigo | Yellow | | | 57° 55' | Sombre purple | White | | | 57° 50' | Sombre reddish violet | White | | | 57° 20' | Dirty violet yellow | White | | | | Pale yellow | Violet white | | | 55° 40' | White | Sombre violet | | | 55° 33' | Pure brilliant white | Black | Virtual poles. | | | White | Sombre dirty green | | | | Pale orange | Pale dirty bluish green | | | 53° 40' | Sombre orange or brick red | White | | | 53° 27' | Sombre and narrow purple | White | | | | Blue | Ruddy white | | | | Pale blue | Orange | | | | Bluish white | Orange red | | | 132° 20' | White | Narrow crimson | | | 51° 3' 132° 40' | Pale yellow, &c. &c. | Purple, &c. &c. | | Here the virtual poles \( p, p' \) correspond precisely to the first minimum of the extraordinary pencil. In a plate of Rochelle salt, cut nearly, but not quite perpendicular to the optic axis, and whose thickness was 0.194425 inches, the rings beyond the poles were almost entirely obliterated, while those between them exhibited the following singular succession of colours, which will show to what an extent the deviation from Newton's scale is carried in this substance. **Table IV. Rochelle salt, perpendicular to the optic axis. Thickness = 0.194425.** | Corresponding inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |---------------------------|-----------------|---------------|-------------| | 201 30 330 10 | White | White | Poles PP' for red rays. | | 207 10 | Exceeding pale blue | Exceeding pale pink | | 209 0 | Exceeding pale pink | Exceeding pale blue | | 210 30 | Very pale blue | Very pale pink | | 212 30 | Very pale pink | Very pale blue | | 213 20 | Very pale blue | Very pale pink | | 215 25 | Pale pink | Pale blue green | | 216 40 | Pale blue | Pale yellow pink | | 218 0 | Pink | Blue green | | 219 29 | Pale blue | Pale yellow pink | | 220 58 | Pink | Greenish blue | | | White | Very pale purple | | 222 30 | Blue | Yellow pink | | 223 45 309 30 | Yellowish pink | Greenish blue | | 224 50 308 15 | Pale greenish blue | Yellowish pink | | | Blue | Pale pinkish yellow | | 226 10 307 0 | Pale pink | Pale greenish blue | | | Pale yellow | Dark blue | | | White | Pale purple | | 227 25 | Bluish or greenish white | Very pale violet pink | | 228 30 | Blue | Very pale yellow | | | Violet almost imperceptible | White | | 229 3 304 10 | Pure white | Pure white | Virtual poles p, p'. | | | Exceeding pale yellow | A little violet | | 229 50 | Pale yellow | Very narrow dark blue | | | Very pale tawny orange | Blue, sombre and pale | | 231 20 302 20 | Fine purplish crimson | Pale yellow green | | | Very pale purple | White | | 232 25 | Very pale green | Fine crimson | | | White | Pale purple | | 234 0 299 20 | Splendid crimson | Very pale green | | | Pale purple | White | | 235 40 | Very pale green | Rich crimson | | | White | Pale purple | | 236 23 296 15 | Rich crimson | Very pale green | | | Pale purple | White | | 238 50 294 35 | Pale blue green | Good pink, almost crimson | | | White | Very pale purple | | 240 12 292 55 | Pink rather pale | Pale greenish blue | | | White | White | | 242 10 291 30 | Pale blue | Pale yellowish pink | | 243 40 289 50 | Pale pink yellow | Pale blue | | 245 10 287 45 | Very pale blue | Very pale yellow | | 246 50 286 45 | Very pale yellow green | Very pale lilac blue | | 248 25 284 55 | Pale lilac | Yellow green | | | White | White | | 250 40 282 30 | Fine yellow green | Fine lilac | | | White | White | | 252 57 280 12 | Lilac | Fine yellow green | | | White | White | | 256 5 277 25 | Fine yellow green | Fine lilac | | | White | White | | 259 20 274 40 | Pale lilac | Fine yellow green | | 261 40 271 40 | White | White | | 266 40 266 40 | Green yellow | Pale lilac | The middle tint | In order however to avoid the effect of the dispersive power, which at such considerable obliquities would render the observations liable to some uncertainty, I cut another plate, in such a manner that the perpendicular to its surface, instead of coinciding nearly with the optic axis, was directed very nearly to one of the virtual poles. Its thickness was then gradually reduced in the manner above described for sulphate of baryta, though, owing to the nature of the body, it was found impossible to avoid the necessity of re-polishing the posterior surface at each operation; but as this was done with all possible care, only a very slight error can have arisen from this cause. **Table V. Rochelle salt. Thickness = 0.11518.** | Inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |--------------|-----------------|---------------|-------------| | 277° 8' | Very pale pink | Very pale bluish green | Pole P for mean red rays. Perpendicular incidence. | | 261° 8' | White tinged with orange | Very fine intense indigo | | | 260° 25' | Yellowish or greenish white | Purple rather sombre | | | 259° 45' | Very pale greenish blue | Indifferent purplish pink | | | 258° 50' | Dull blue | Yellowish pink white | | | 258° 30' | Fine deep indigo | White inclining to orange | | | 257° 35' | Violet purple | Yellowish white | | | 257° 35' | White, a little tinged with violet | White, not very brilliant | The virtual pole p. | | | Yellowish white | Pale violet blue | | | 256° 30' | Pale yellow | Sombre indigo inclining to violet, narrow & well defined. | | | | Pale pink yellow | Sombre violet white | | | 255° 10' | Pinkish purple | Very pale greenish yellow | | | 254° 30' | Rich sombre purple somewhat fiery | White tinged with greenish yellow | | | | Pale green | Fine crimson | Flow | | 252° 30' | Extremely pale green | The richest deep damask crimson | | | | White | Livid imperfect purple | | | 250° 27' | The richest damask crimson | Fine pale green | | | | Livid imperfect purple | White | | | 248° 5' | Pale bluish green | Fine rich crimson | | | 245° 45' | Pink approaching to red | Pale blue green | | | 243° 30' | Sky blue | Light pink strongly inclining to pale greenish blue [orange red] | | | 240° 25' | Pink orange | Fine yellow | | | | Sky blue inclining to lilac | Fine pink a little purple | | | 237° 30' | Pale bluish green | Lilac | | | 237° 0' | Splendid yellow green | Splendid green | | | 234° 20' | Rich lilac | Rich lilac | | | 230° 10' | Splendid green | White | | | | White | Splendid yellow green | | | 225° 45' | Lilac blue | White | | | | White | Pale lilac blue | | | 220° 40' | Splendid green yellow | White | | | | White | Pale greenish yellow | | | 211° 40' | Pale lilac | Fine lilac | | | 197° 30' | Fine yellow green | | Here the virtual pole was coincident with the 5th maximum (or thereabouts) of the extraordinary ray from the pole P; the succession of tints, however, unless close to the virtual pole, is omitted, in order to shorten the table. **Table VI. Rochelle salt. Thickness = 0.08557.** | Inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |--------------|-----------------|---------------|--------------| | 262 | Good light pink | Light blue green | Pole P for mean red. Perpendicular incidence. | | 278 | Very pale yellow green | Bluish purple | | | 278 20 | Very pale yellow green | Violet | | | 278 55 | Very pale bluish green | Very light pink, or pinkish white | | | 280 | Indigo | Very pale yellow, almost white | | | 281 30 | Pale violet | Very pale yellow | | | 281 30 | White | White, perfectly equal and alike Virtual pole. | | | 282 35 | Very pale greenish yellow | Pale lilac | | | 282 35 | Pale greenish yellow | Sombre lilac purple | | | 282 35 | Pale pinkish yellow | Dull and impure blue | | | 285 15 | Pale pink | Pale yellow green | | | 285 45 | Crimson | Yellow green | | | 285 45 | Rich fiery damask crimson | Very pale yellow green | | | 287 50 | Livid imperfect purple | Very pale pink yellow | | | 288 45 | Fine light green | Fine pink | | | 288 45 | Very pale green | Splendid crimson | | | 291 40 | Good crimson | Blue green | | | 291 40 | Crimson, almost scarlet, &c. | Pale blue green, &c. | | In this plate the virtual pole fell about half way between the 4th maximum and the 5th minimum of the extraordinary ray from the apparent pole P for the mean red rays. When once more ground down, it gave as follows: ### Table VII. Rochelle salt. Thickness = 0.05437. | Inclinations | Ordinary pencil | Extraordinary | Remarks, &c. | |--------------|-----------------|---------------|--------------| | 262 25 | Fine pink | Fine light blue green | PoleP for mean red rays. | | | Indifferent purple | Yellowish white | | | 278 20 | Indifferent lilac pink | Very pale greenish yellow | Perpendicular incidence. | | 278 25 | Pale yellow inclining to orange | Blue rather pale | | | 280 50 | Fine pale yellow | Beautiful sombre indigo | | | 282 0 | Yellowish white or pale yellow | Violet | | | 282 0 | White | White with an almost imperceptible tinge between yellow | Virtual pole | | | | Yellow white [and violet] | | | 284 50 | Very pale blue | Pale yellow white | | | | Sombre indigo | Extremely pale pink white | | | | Very pale blue | Lilac pink | | | | Yellow green | Deep lilac pink | | | 288 25 | Pale yellow green | Rich but sombre purplish crimson | | | | Greenish white | Dull purple [son] | | | 293 33 | White | Good blue green | | | | Very pale pink | White | | | | Deep fiery crimson | White | | | | Very dull purple (greenish) | White | | | | Blue | Pink yellow [&c.] | | | 294 50 | Very pale blue, &c. | Rich orange, bordering on red, | | ### III. On the causes of these phenomena. The development of colour along the axis of double refraction, is at first sight analogous to the production of the secondary tints along the axis of rock crystal, discovered by M. Arago, and recently explained by M. Biot, in a masterly memoir communicated to the Academy of Sciences, on the hypothesis of a force inherent in its molecules independent of their state of aggregation, by which they communicate a rotation in an invariable direction to the axes of polarisation of the luminous rays. And this analogy is partially supported by the fact, that the tint developed along the axis, descends in the scale of colour as the thickness increases. A more scrupulous examination however will show, that its origin must not be sought in any cause of this nature, for (not to mention the impossibility of explaining the phenomena of the virtual poles by this hypothesis) if we place the principal section of the crystal in the azimuth zero, the extraordinary image will be found to vanish completely for every angle of incidence, and whatever be the thickness of the plate. I may add too, that I have in my possession a crystal of quartz, which exhibits with tolerable distinctness in some parts the phenomena of two axes, and the appearances produced by the interference of the secondary tints in this specimen, while they agree completely with M. Biot's explanation, differ entirely from those which form the subject of this Paper. Neither are the phenomena above described explicable on any supposition of a peculiar action of the crystal on the differently coloured rays, analogous to its ordinary or extraordinary dispersive power, by which the periods of alternate polarisation of the molecules of some colours, should be lengthened, and of others contracted, so as to disturb that exact proportionality to their periods of easy reflection and transmission, which M. Biot has proved to be a necessary condition for the production of the tints of Newton's scale. It is true, such laws of action may be imagined, and I shall presently show must really exist; in all crystals probably to a small extent, but in two instances at least, to a surprising degree. But this alone will avail us nothing. To show this, and at the same time obtain a general analytical expression for the tint developed at any inclination, and for every hypothesis of the action of the crystal on the differently coloured molecules, let us denote by $c$, the length of a complete period of easy transmission and reflection, or the extent of one pulse, on the undulatory hypothesis, in vacuo, and at a perpendicular incidence for any homogeneous ray, and let C denote its colour and proportional intensity or illuminating power in the prismatic spectrum. Then will the formula representing a beam of white light intromitted into the crystal, be \[ C + C' + C'' + \&c. \] from one end of the spectrum to the other. Let \( n \) be the number of periods (each consisting of a double alternation) and parts of a period performed by the elementary pencil C, in its passage through the medium: then, according to the theory of M. Biot, when \( n \) is 0, 1, 2, 3, &c. ad inf. the pencil will wholly pass into the ordinary image; but when the values of \( n \) are \( \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \&c. \) it will wholly* be thrown into the extraordinary one, and in the intermediate states of \( n \), partly into one and partly into the other. These conditions are satisfied if we represent by \( \sin^2(n\pi) \) the intensity of the ray in the ordinary image, taking unity for its original intensity; and it will I believe be found, that the gradation of intensity given by this formula for the inter- --- * The amplitude or total extent of each oscillation of the plane of polarisation is here supposed 90°, in which case the contrast of colour in the two pencils is at its maximum. This is the case in the situation we are considering, but in general the intensity of the extraordinary ray, instead of being represented by \( \sin^2(n\pi) \), will have an additional factor, a function of the azimuth A of the principal section of the crystallized plate and the position of the refracted ray, and which becomes unity when \( A = 45^\circ \), and the plane of incidence is that of the principal section. It is on this factor that the gradation of brightness in the isochromatic lines, and the black cross or hyperbolic branches which intersect them, depend. But it is not my intention at present to enter on this part of the subject for reasons to be explained farther on. See note in page 84. mediate values of $n$, will agree sufficiently with the judgment of the eye to warrant its adoption.* The part of the elementary pencil $C$ then, which enters into the extraordinary image, will be $C \cdot \sin^2 (n \pi)$. Let us denote by $S \{ C \cdot \sin^2 (n \pi) \}$ the aggregate of all such elements from one extremity of the spectrum to the other, or take $$S \{ C \cdot \sin^2 (n \pi) \} = C \cdot \sin^2 (n \pi) + C' \cdot \sin^2 (n' \pi) + \&c.$$ Then will this expression represent the tint developed in the extraordinary image, and consequently, $S \{ C \cdot \cos^2 (n \pi) \}$ that in the ordinary one. Now, $n$, the number of periods performed depends, 1st. on the nature of the ray, or on $c$; 2dly, on the intrinsic energy of the action of the medium on that ray; and 3dly, on the direction of its course, the thickness of the plate, and whatever other cause or limit of periodicity may happen to prevail. Hence we may take $n = M \times k$, $k$ being a function of $c$, dependent only on the nature of the body through which the ray $C$ passes, and $M$ being a certain multiplier whose form we shall consider presently. This substitution made, the expression for the tint becomes $S \{ C \cdot \sin^2 (Mk \cdot \pi) \}$ In the theory of the Newtonian colours of thin plates and the polarised rings in crystals with one axis, the multiplier $M$ is independent on $c$, varying only with the direction of the ray and the thickness of the plate. It is therefore the same for all the coloured rays, and the tint, for any value of $M$, will be * No part of our subsequent reasoning depends on the form of this function. It is sufficient to know that it must be a periodical, and even function of $n$. It is only in the computation of numerical values that it is necessary to make any more precise assumption. MDCCCXX. Mr. J. F. W. Herschel on the action of \[ C \sin^2 (M k \pi) + C' \sin^2 (M k' \pi) + \&c. \quad (a) \] Now, suppose \( M \) to begin from zero, and to pass, by a variation either in the direction of the ray or thickness of the medium, or both, through all gradations of value, to infinity, or to its maximum, if not susceptible of infinite increase: then we see that for every value of \( M \) a certain peculiar tint will arise, and that, provided \( M \) commence at zero and continue increasing, the same succession of tints will invariably be developed in the same order. Consequently, if we fix upon any two tints in this scale of colour, or any two values of \( M \), the same succession and the same number of alternations of colour must invariably intervene between them, however we pass from one to the other. In a crystal with two or more axes, the value of \( M \) for any ray \( C \) must of course be zero in the direction of the axis, and therefore if the same supposition of the independence of \( M \) on \( c \) be made, the same conclusions should follow; namely, 1st. that the extraordinary ray must always vanish in the pole, whatever be the thickness of the plate; and 2dly, that the same succession and number of alternations of colour should intervene between the pole and any assigned unequivocal tint, such as black, or the pure brilliant green of the 3d order of Newton's scale. Both these conclusions are totally at variance with the facts above detailed, as to the development of colour in the poles, and the situation in the order of the rings of what we have called the virtual poles. Hence we are necessitated either to give up the theory of alternate polarisation altogether, or to admit the dependence of the multiplier \( M \) on \( c \), or on the nature of the ray. Let us see to what this will lead us. According to the theory of the polarised rings if extended to crystals with two axes, the number of periods performed in a given space (=1) by a molecule of a given colour, transmitted in a direction making angles $\theta, \theta'$ with the axes, can only be a function of the form $k \cdot \psi(\theta, \theta')$, $k$ depending on the intensity of the polarising force; or as before, being a function of $c$, the nature of the ray, and of the intrinsic energy of the molecules of the crystal. Now if we call $t$ the thickness of the plate, and $\phi$ the angle of refraction, $\frac{t}{\cos \phi}$ is the length of the path described, and therefore we must have for the number of periods $$n = \frac{k t}{\cos \phi} \cdot \psi(\theta, \theta');$$ so that the value of $M$ must be $\frac{t \cdot \psi(\theta, \theta')}{\cos \phi}$, which must be a function of $c$. Now $t$ is obviously independent of it; and if we neglect at present the very trifling effect at moderate incidences of the ordinary dispersive powers of the media examined,* $\phi$ is so also. It is therefore in the form of the function $\psi(\theta, \theta')$ that we must look for the cause of the phenomena; and since, we have $\theta' = \theta + 2a$, $2a$ being the angle between the axes (because the observations are made in the principal section) we see that $\psi(\theta, \theta + 2a)$ must involve $c$, and consequently, $\theta$ being arbitrary and independent, $a$ must be a function of $c$. In order then to render the theory of alternations applicable, we must admit the angle between the * It is easy to see that in the two classes of crystals above described, the effects of the dispersive powers will be opposite to each other, in one opposing, and in the other conspiring with the causes which produce the deviation of tints. In the tables, Nos. V, VI, VII, where the virtual poles were observed almost at a perpendicular incidence, the influence of the dispersive power is quite insensible. axes of double refraction to differ in the same crystal for the differently coloured rays. We must now show that this supposition is sufficient to represent the phenomena correctly. The symmetry of the rings and total evanescence of colour in the principal section at an azimuth zero, requires that the axes of all the different colours shall be symmetrically arranged, on either side of a fixed line (which may be called the optic axis) in this plane, or in one perpendicular to it. At present we need only consider the former case. Let \(a\) represent the angular distance of the axis for any one standard species of ray C (the extreme red, for instance) from this line, \(a + \delta a\), the same distance for any other ray. Then the distance of the transmitted ray C, from the axes of rays of that colour being \(\theta, \theta'\), the corresponding distances from their respective axes for rays of any other colour C' emerging in the same direction will be \(\theta - \delta a + \delta \phi\) and \(\theta' + \delta a + \delta \phi\), \(\delta \phi\) being the difference \((\phi' - \phi)\) of the angles of refraction, corresponding to the same incidence, for the colours C, C'. The positive values of \(\theta\) here reckon outwards from the pole; \(\delta a\) is negative for crystals of the second class, and \(\delta \phi\) is negative or positive according as C or C' is the less refrangible colour. Let us for a moment consider rays of only these two colours. The portion of the extraordinary pencil due to them will be \[C \cdot \sin^2 \left( \frac{k t}{\cos \phi} \psi (\theta, \theta') \cdot \pi \right) + C' \cdot \sin^2 \left( \frac{k' t}{\cos \phi'} \psi (\theta - \delta a + \delta \phi, \theta' + \delta a + \delta \phi) \cdot \pi \right).\] The rays of these colours of the same order in their respective series of rings will therefore coincide, and that in the proper degree of proportional intensity for the production of a white image, provided we suppose \[ \frac{k}{\cos \varphi} \cdot \psi(\theta, \theta') = \frac{k'}{\cos \varphi'} \cdot \psi(\theta - \delta a + \delta \varphi, \theta' + \delta a + \delta \varphi); \quad (b) \] which, since \(k, k', a, \delta a\) are constant elements, \(\varphi, \varphi'\) determinate functions of \(\theta\), and \(\theta' = \theta + 2a\), suffices to determine \(\theta\). If we suppose C and C' to represent the extreme red and violet rays, it is evident that the coincidence of the extraordinary pencils of the same order for these two extremes, will ensure that of the intermediate ones, at least very nearly. It would do so precisely, were the value of \(\delta a\) for any intermediate ray, such a function of \(k\) as would result from making \(\theta\) constant in the preceding equation, because the two laws, that of the dispersion of the axes, and that of the magnitude of the rings of different colours, would then act in exact opposition to each other throughout their whole extent. It is, in fact, a case precisely analogous to that of the compound achromatic prism, where if the law of dispersion in the one medium were identical with that in the other, a perfectly colourless pencil would emerge, and when these laws differ, the coincidence of the red and violet rays ensures an approximate coincidence of all the rest. Should these laws however differ very considerably, an uncorrected colour will appear at the point so determined, and a nearer approximation will be obtained by uniting two of the more powerful intermediate rays, such as for instance the mean red and the blue, or limit of the green and blue. This then is the origin of the virtual poles or points beyond or between the axes where the tint rises to a white of the first order, more or less feeble, or even to an absolute black; and we may now see the reason why the tints in reckoning from these points approximate in a general way to the Newtonian scale. In fact, the periods of the more refrangible rays being performed more rapidly than those of the less, if we suppose the coincidence above spoken of to take place at any point (the minimum for instance), of the \( n^{th} \) ring, the intervals between the \( n^{th} \) and \((n \pm 1)^{th}\) minimum will be greatest for the red and least for the violet, &c. Consequently, when the violet next disappears totally from the extraordinary pencil, there will remain yet a little of the red, less of the orange, and so on, and this difference increasing at every succeeding minimum on either side, will produce a succession of colours approximating in a general way to Newton's scale. This approximation will however be much less close on the side of the virtual pole towards the nearest axis, because the disturbing influence of the separation of the axes on the figure of the rings and the law of their successive intervals, is much more sensible than at a distance from the pole. This will be evident if we consider that in the interval between the extreme coloured axes, the tints will be regulated entirely by the law of their distribution. Now this is perfectly corroborated by the succession of tints in the foregoing tables, as well as by numerous experiments made on other bodies. Our equation \((b)\) gives room for a remark of some consequence, as it affords a striking verification of the theory here delivered. It will be observed that this equation does not involve \( t \), and in consequence, the angle \( \theta \) determined from it, at which the coincidence takes place, is the same for all values of \( t \), or for all thicknesses of the plate. The observations of the tints in the tables given above, afford us ample means of putting this remarkable consequence to the test of experiment. In fact, in the three series of tints observed in sulphate of baryta, the apparent angles between the axes for the mean red rays are respectively $62^\circ 0'$, $62^\circ 2'$, and $61^\circ 53'$, the mean of which is $61^\circ 58'$, while the apparent angles between the virtual poles in the same series are $72^\circ 46'$, $72^\circ 36'$, and $72^\circ 47'$. The semi-excesses of the latter angles over the mean value of the former, are the apparent angular distances of the virtual poles from the axes of mean red rays, and are respectively $5^\circ 24'$, $5^\circ 19'$, $5^\circ 25'$, neither of which differs more than $4'$ from the mean. To ascertain the real value of $\theta$, we have only to compute the angles of refraction. In the specimen employed, I found $1.6475$ for the index of ordinary refraction, and the angles of incidence (the halves of the above angles) being $30^\circ 59'$ and $36^\circ 23'$, $36^\circ 18'$, $36^\circ 24'$; the corresponding angles of refraction are $18^\circ 12'$ $30'' (= a)$, $21^\circ 6'$ $10''$, $21^\circ 3'$ $40''$, $21^\circ 6'$ $40''$; and since $\theta = \phi - a$ in this case, we find for the values of $\theta$, $2^\circ 53'$ $40''$, $2^\circ 51'$ $10''$, $2^\circ 54'$ $10''$, the mean of which gives $2^\circ 53'$ $0''$ for the real angular distance of the virtual pole from the axis of mean red rays in sulphate of baryta. Again, in the series of tints, tables V, VI, VII, for Rochelle salt, the apparent angular distances of the mean red axis from the virtual pole were $277^\circ 8' - 257^\circ 35' = 19^\circ 38'$; $281^\circ 30' - 262^\circ 0' = 19^\circ 30'$, and $282^\circ 0' - 262^\circ 25' = 19^\circ 35'$ of which neither differs more than $3'$ from the mean $19^\circ 38'$. Dr. Brewster (Phil. Trans. 1814, p. 216) has stated the refractive index of this salt at $1.515$; but this is certainly a little too large. In four experiments made at distant intervals of time, and by different modes of observation, I have found \[ 1.49640 \quad 1.50220 \\ 1.49670 \quad 1.49853 \] for the index for the mean yellow-green rays, of which determinations the last is to be preferred, having been made with great care. The same experiment gave \(1.49293\) for the index for mean red rays. The apparent angular distance of the axis for red rays from the perpendicular was \(16^\circ\), which leaves \(3^\circ 33'\) for the angular distance of the virtual pole from the perpendicular. These angles of incidence correspond to the respective angles of refraction \(10^\circ 38' 20''\) and \(2^\circ 22' 40''\), of which the sum \(13^\circ 1'\) is the real angle between the virtual pole and mean red axis in Rochelle salt. The series in table IV, gives \(13^\circ 2' 25''\) for the value of \(\theta\), which agrees completely with the foregoing determination. I took seven plates of nitre of various thicknesses, and cut from different crystals, and by a mode of observation to be described hereafter, found as follows: **Table VIII.** | Designation of the plate | Mean distance of the virtual poles from the axis of mean yellow rays (apparent.) | Excess above the mean. | Order of the coincidence. | |--------------------------|---------------------------------------------------------------------------------|------------------------|---------------------------| | 1 | \(1^\circ 50'\) | \(-4'\) | Between the 4th minimum and 5th maximum. | | 2 | \(1^\circ 51'\) | \(-3'\) | A little beyond the 3d maximum. | | 3 | \(1^\circ 57'\) | \(+3'\) | Different at the two extremities. | | 4 | \(1^\circ 53'\) | \(0'\) | 2d maximum. | | 5 | \(1^\circ 55'\) | \(+2'\) | 2d minimum. | | 6 | \(1^\circ 59'\) | \(+5'\) | 9th minimum. | | 7 | \(1^\circ 53'\) | \(0'\) | A little beyond the 8th minimum. | | Mean | \(1^\circ 53'\) | | | Although the constancy of the position of the virtual pole for different thicknesses is sufficiently made out here, the small differences which exist are certainly not attributable to errors of observation, which, in the method I employed, are usually confined within much narrower limits. They are due to minute irregularities in the crystals themselves, consisting, probably, in a state of imperfect equilibrium of the molecular forces of aggregation, to which this salt is so subject, that it is rather rare to find a specimen in which the rings beyond both poles have exactly the same breadth or tints. Art. IV. Of the tints developed by crystals with two axes out of the principal section. If we place a crystallized plate at an azimuth zero in a tourmaline apparatus, having the axes of the tourmalines at right angles, we shall observe, if its thickness be at all considerable, that the two oval spots on either side the axis of symmetry (which is now perfectly black) instead of being exactly regular in their figure, as in Fig. 2. Pl. V, and tinged with colours symmetrically disposed on either side of a line \( mn \) perpendicular to the principal section, are invariably coloured at one extremity \( r \) with a strong prismatic red hue, and drawn out at the other \( v \) into more or less elongated and tapering spectra or tails of blue and violet light. The extremities \( r, r \) of the rings too have a large excess of the red rays, and the opposite \( v, v \) of the violet rays. In crystals of the first class above described, the red extremity is turned towards the other pole, while in those of the second it is directed from it. If we subject a plate of Rochelle salt to this examination, the ovals \(a, a\), are drawn out to a surprising length, and the whole prismatic spectrum is displayed in them with great vividness of colour, while the violet portions of the rings are greatly elongated also, and appear to run into one another. If the plate be turned round in azimuth, the phenomena assume the most singular appearances of distortion; and as the rotation approaches to 45°, the rings in the vicinity of the pole are gradually obliterated by their mutual overlapping, which is the greater the thicker the plate. In all situations, however, the interposition of a red or dark green glass immediately restores the perfect symmetry and regularity of the rings, which are then seen in much greater number, and completely well defined. All this is the necessary consequence of the want of coincidence of the axes for different colours. The lateral spots, for example, are formed for each homogeneous colour with perfect regularity close to their corresponding pole, and regularly decreasing in size from the red to the violet. Their arrangement will therefore be as represented in Fig. 3, Pl.V, R, O, Y, &c. being the poles corresponding to the several colours red, orange, &c. The oval spots composed of red rays being represented by \(r, r\), those of the other colours will be super-imposed on them in their order, overlapping, as represented by the dotted ovals \(oo, yy, &c.\) like the circular coloured images of the sun in the spectrum of an ordinary prism, giving rise to the long prismatic tails above described. Similar considerations will apply to the anomalous appearances presented by the rings of all the other orders and in every situation. This suggests a very simple and pleasing experiment, which affords an ocular demonstration of the adequacy of the explanation I have advanced. Let a plate of Rochelle salt be placed in a tourmaline apparatus in any azimuth (45° is the most convenient) and firmly fixed on a proper stand in a dark room. The eye being now applied, let an assistant illuminate the emeried glass or lens of short focus* which disperses the light previous to its incidence on the first tourmaline, with the several colours of the prismatic spectrum in succession, beginning with the red. The rings will then be seen formed successively of each of the colours, perfectly regular in their figure, but contracting rapidly in dimension as they become illuminated with the more refrangible rays.† At the same time the pole about which they form will be seen to move regularly in the direction of the other axis of the crystal, and if we pass alternately from a red to a violet illumination, will shift its position accordingly, backwards and forwards through a very considerable angle. If rays of two colours be thrown at once on the apparatus, their two corresponding sets of rings will be seen at once, crossing, but not obliterating one another, and the distance between their respective centres will be observed to increase with the difference of their colours. By measuring the interval occupied by the projections of * See the description of an apparatus of this kind, subjoined. † See Lectiones Opticae, lib. ii. Pars. i. Obs. xiii. from which the idea of this experiment is taken. "Magnâque voluptate perfusus" says Newton, with the enthusiasm of the true philosopher who loves the field he labours in, "videbam eos dilatare aut contrahere se gradatim pro eo ac color luminis immutabatur." It is impossible to witness the very beautiful phenomenon described in the text without entering into the same train of feeling. the centres of the last visible red and violet rings, as well as those of the intermediate colours, on a screen at a known distance seen with the other eye, I found the following values of the apparent and real separation of the several coloured axes: | Between | Apparent interval | Real Interval | |---------------|-------------------|---------------| | Red and orange| 0° 37 very uncertain | 0° 25 | | Yellow | 1° 50 rather uncertain | 1° 13 | | Green | 3° 43 Do. | 2° 29 | | Blue | 6° 5 | 4° 3 | | Indigo | 8° 19 | 5° 33 | | Violet | 9° 46 | 6° 31 | As a mode of measurement this method is very inaccurate, especially in the extreme red and violet rays, both of which would be copiously, and indeed almost entirely absorbed in their passage through two plates of tourmaline of a yellowish-green colour. Much more exact and unexceptionable measures will be presently given, but these are quite sufficient to establish the reality of the phenomenon described. V. Of a secondary cause of the deviation of tints, subsisting in certain crystals, and of the anomalous tints of the apophyllite. To determine the dispersive power of any medium, and obtain some rough knowledge of its law, we make a prism of it act in opposition to one of a standard substance. To ascertain the total dispersion of the axes of a crystal, or the angle by which the extreme red and violet axes differ, we may make it act against itself. Since the violet rings are more elevated by refraction than the red, from the situation in which they would appear to an eye immersed in the medium, a plate may be conceived cut in such a direction as to make their apparent centres coincide, in which case the tints immediately about the poles will coincide with Newton's scale, and the extraordinary image will totally disappear in the pole at an azimuth $45^\circ$. This condition gives $\theta = 0$, $\theta - \delta a + \delta \phi = 0$, whence (supposing $R, R'$ the indices of refraction for extreme red and violet rays and $\delta R = R' - R$) we find $$\delta a = \delta \phi = \frac{\delta R}{R} \tan \phi$$ The angle $\phi$ however becomes imaginary, and this method, in consequence, inapplicable when the separation of the extreme axes ($\delta a$) is greater than the maximum dispersion of the colours of an intromitted white ray, that is, when $$\delta a > \frac{\delta R}{R \sqrt{R^2 - 1}}$$ Let us resume our equation $(b)$, and supposing the form of the function $\psi$, and the constants $a, k, k', R$ and $\delta R$ ascertained, let the angle $\theta$, at which the coincidence takes place be observed, and the value of $\delta a$ will then become known. If we suppose it small, which it is in the generality of crystals, we get $$\delta a = \frac{k - k'}{k'} \psi - \delta \phi \left\{ \frac{k}{k'} \sin \phi \cdot \psi + \frac{d \psi}{d \theta} + \frac{d \psi}{d \phi} \right\}$$ $$\frac{d \psi}{d \theta} - \frac{d \psi}{d \phi}$$ $(c)$ ($\psi$ being put for $\psi(\theta, \theta')$ for the sake of brevity). At incidences nearly perpendicular, $\delta \phi$ may be neglected, and the expression reduces itself to $$\delta a = \frac{k - k'}{k'} \frac{\psi}{\frac{d \psi}{d \theta} - \frac{d \psi}{d \phi}}$$ $(d)$ The comparison of these formulae with observation, which will lead to some very remarkable consequences, requires us to know the form of the function $\psi$ and the values of $k, k'$. We will begin with the former, and in this investigation the first step is to determine the general equation of the isochromatic lines. In order to this, we must separate in all cases the law of the tint from that of its intensity. The latter depends entirely on the greater or less facility with which the emergent ray finds in penetrating the prism of Iceland spar employed for its analysis, and will not enter into the present investigation. When we examine a crystallized plate in a convenient graduated apparatus, between tourmaline plates crossed at right angles, turning it slowly round between them in its own plane, the form of the coloured bands, if illuminated with homogeneous light, will remain perfectly unchanged during the rotation, but the two black hyperbolic branches passing through the poles, will obliterate in succession every part of their periphery; and the space over which the darkness extends, as well as the degree of illumination of what remains visible, varying at every instant, give rise to so great a variety of appearances, that some little attention is required to recognize this perfect identity of figure. When the tourmaline next the eye is made to revolve, the crystallized plate remaining fixed, the complicated changes which take place, are perfectly reconcilable with the superposition of the primary on its complementary set of rings, the relative intensities of the two sets at any point being regulated by laws we have no occasion to consider at present, but the figure of the isochromatic lines, where visible, remains absolutely unchanged by any rotation in this part of the apparatus. To form a first hypothesis on the nature of the function which determines the equation of any one of these curves, we must select a crystal, where the proximity of the axes and intensity of the polarising forces are such, as to bring the whole system of rings within a very small angular compass; as by this means we avoid almost entirely the disturbing effect of the variation in thickness, arising from obliquity of incidence. Dr. Brewster, in his Paper of 1818, has chosen nitre, as affording the best general view of the phenomena, and it is admirably adapted for this purpose; the whole system of rings being comprised at a very moderate thickness within a space of $10^\circ$, allowing us to regard their projection on a plane perpendicular to the optic axis as a true representation of their figure, undistorted by refraction at the surface, &c. If we examine the rings in this crystal (illuminated with homogeneous light, or by the intervention of a red glass in common day-light) it will be evident that the general form of any one of them is a re-entering symmetrical oval, which no straight line can cut in more than four points, and which, by a variation of some constant parameter, is in one state wholly concave, as 1 (Fig. 4. Plate V.) then becomes flattened, as 2; then acquires a minimum ordinate and points of contrary flexure, as 3; then a node, as 4; after which it separates into two conjugate ovals, as 5; which ultimately contract themselves into the poles P, P' as conjugate points. The general idea bears a striking resemblance to the variation in form of the curve of the fourth order, so well known to geometers under the name of the lemniscate, whose equation is $$(x^2 + y^2 + a^2)^2 = a^2 (b^2 + 4x^2)$$ when the parameter $b$ gradually diminishes from infinity to zero, \(2a\) being the constant distance between the poles. In order, however, to put this to a satisfactory examination, accurate measures must be taken, which, in the case of nitre, from the minuteness of the system of rings, presented at first some difficulties. These I obviated, after many fruitless trials, by a mode of observation which I have found extremely convenient and accurate, and which applies particularly well to the present purpose. It consists in projecting the rings by solar light on a screen in a darkened room, by which means they may be magnified to any required extent, examined at perfect leisure, and in all their phases, and measured or traced with a pencil with the utmost exactness and facility. They may be thus exhibited too to a number of spectators at once; a thing which may prove serviceable to the Lecturer, for which reason I have subjoined to this paper a brief description of the apparatus I employ. Having cut a very perfect crystal of nitre at right angles to its axis of crystallization, and adjusted it properly on this apparatus, the rings were projected on a large sheet of paper, stretched, while moist, on a drawing board, by which means it assumes a truly plane surface by the contraction it undergoes while drying. The poles were then marked, and the loci of the successive maxima for red rays carefully outlined. The screen being then removed, a series of lemniscates were laid down by points, having the same poles and one common point in each, chosen where the tint was most decided. It is unnecessary to give any comparative statement of measures in the observed and constructed curves, as the points, graphically laid down, uniformly fell on the pencilled outlines, or, in the few instances to the contrary, within limits less than the very trifling irregularities of the outlines themselves. The graphical construction of these curves is rendered extremely easy by the elegant and well-known property of the lemniscate, in which the rectangle under two lines drawn from the foci (or poles) to any point in the periphery, is invariable throughout the whole extent of the curve. This is easily shown from its equation, and the value of this constant rectangle in any one curve is expressed by $a \times b$. We must next enquire how the constant parameter $b$ varies in passing from ring to ring. To this end I projected the rings, illuminated by red light only, on a screen as before, and having outlined the successive loci of the minima of illumination, and laid down the poles, found the values of $ab$ in the several lemniscates, as in the following table: | Order of the minimum. | Observed values of $ab$ in square inches. | Differences. | Values of $ab$ computed from formula $ab = 1.59 \times n$ | Excess of computed above observed values of $ab$ | |----------------------|------------------------------------------|--------------|-------------------------------------------------|-----------------------------------------------| | $n = 0$ | $0.00$ | | $0.00$ | $0.00$ | | 1 | $1.62$ | $1.62$ | $1.59$ | $-0.03$ | | 2 | $3.165$ | $1.545$ | $3.18$ | $+0.02$ | | 3 | $4.69$ | $1.525$ | $4.77$ | $+0.08$ | | 4 | $6.27$ | $1.58$ | $6.36$ | $+0.09$ | | 5 | $7.87$ | $1.60$ | $7.95$ | $+0.08$ | | 6 | $9.56$ | $1.69$ | $9.54$ | $-0.02$ | | 7 | $11.09$ | $1.53$ | $11.13$ | $+0.04$ | | 8 | $12.77$ | $1.68$ | $12.72$ | $-0.05$ | | 9 | $14.33$ | $1.56$ | $14.31$ | $-0.02$ | | 10 | $15.93$ | $1.60$ | $15.90$ | $-0.03$ | Mean $1.59$ The nature of the illumination not allowing the delineation to be performed with the same freedom and precision as in a fuller light, the values of $ab$ in the second column are the means of a great number of measures, taken in every part of their respective curves. The numbers in the 5th column exhibit the excesses of the terms of the arithmetical progression in the 4th (whose common difference is $1.59$, the mean of all the differences in the third column) above the observed values of $a b$, and are so small as fully to authorize the conclusion, that these values, and of course those of the parameter $b$, increase in arithmetical progression with the order of the rings; or in other words, that the number of periods performed in a given space ($=1$) by a luminous molecule going to form any point $M$ in the projection of any ring, is proportional to the rectangle of the distances $P M, P' M$ of that point from the two poles. Now, if we extend our views to crystals in which the distance between the axes is considerable, we may reasonably expect that the usual transition which takes place in analytical formulae from the arc to its sine, when we pass from a plane to a spherical surface, will hold good. If this be the case, we shall have at once, and in all cases $$\psi(\theta, \theta') = \sin \theta \cdot \sin \theta'$$ and the nature of the isochromatic curve for the $n^{th}$ complete period will be expressed by the equation $$\sin \theta \cdot \sin \theta' = \frac{n}{k t} \cdot \cos \phi = n h \cdot \cos \phi$$ (e) putting $h$ for $\frac{1}{k t}$. If the plate be cut at right angles to the optic axis $$\cos \phi = \frac{\cos \theta + \cos \theta'}{2 \cdot \cos a}$$ and consequently $$\sin \theta \cdot \sin \theta' = \frac{n}{2 k t \cdot \cos a} (\cos \theta + \cos \theta');$$ (f) To put this to the trial, I took a plate of mica, whose thickness measured $0.023078$ inch, and having adjusted it accurately on a divided apparatus, placed it in an azimuth $45^\circ$, and, by the intervention of the red glass above mentioned, observed the maxima and minima of the extraordinary pencil between the poles. As these observations, when repeated, seldom agreed unless within a few minutes, ten were taken of each maximum and minimum. The angles of incidence, deduced from a mean of similar observations on each side of the perpendicular, are set down in the 2d column of the following table, each number in which is therefore a mean result of 20 observations. The 1st column contains the values of $n$, or the order of the ring observed; the 3d, the angles of refraction, to obtain which I used the index 1.500, employed by M. Biot.* The 4th and 5th columns contain the values of $\theta$, $\theta'$ thence computed, and the 6th, values of the coefficient $h$, deduced from the formula $h = \frac{\sin \theta \cdot \sin \theta'}{n \cdot \cos \phi}$ | Values of $n$ | Angles of Incidence | Angles of Refraction | Values of $\theta$ | Values of $\theta'$ | Values of $h$ | Excesses above mean | |---------------|---------------------|---------------------|-------------------|-------------------|--------------|-------------------| | 0 | 35° 3' 30" | 22° 31" | 0° 0' 0" | 45° 2' 0" | | | | 0.5 | 32° 55' 20" | 21° 14' 40" | 1° 16' 20" | 43° 45' 40" | 0.032952 | -0.000195 | | 1 | 30° 34' 40" | 19° 49' 30" | 2° 41' 30" | 42° 20' 30" | 0.033622 | +0.000475 | | 1.5 | 28° 15' 20" | 18° 24' 0" | 4° 7' 0" | 40° 55' 0" | 0.033035 | -0.000112 | | 2 | 25° 34' 20" | 16° 43' 30" | 5° 47' 30" | 39° 14' 30" | 0.033327 | +0.000180 | | 2.5 | 22° 46' 20" | 14° 57' 15" | 7° 33' 45" | 37° 28' 15" | 0.033148 | +0.000001 | | 3 | 19° 35' 40" | 12° 55' 10" | 9° 35' 50" | 35° 26' 10" | 0.033058 | -0.000089 | | 3.5 | 15° 48' 40" | 10° 27' 50" | 12° 3' 10" | 32° 58' 50" | 0.033026 | -0.000121 | | 4 | 10° 48' 50" | 7° 11' 10" | 15° 19' 50" | 29° 42' 10" | 0.033010 | -0.000137 | * Recherches sur les Mouvements des Molécules de la Lumière, &c. p. 482. He takes it equal to that of glass—"ce qui ne doit pas être fort éloigné de la vérité." I have attempted, without success, to measure its value. What has satisfied M. Biot and Dr. Brewster (for the latter has evidently used this index, or one very near it, Phil. Trans. 1818, p. 230) ought to satisfy every one; yet it is fortunate that in the present instance, a slight variation in the refractive index will produce but a very trifling change in the relative values of $b$. The last column of this table exhibits the deviations in excess or defect of the values of the quantity $h$, so computed from the mean of all of them. Their smallness, in comparison with the quantity itself, and their alternations of sign, are evident proofs of the constancy of this coefficient, and we are therefore authorized to take $\sin \theta \times \sin \theta'$ as the general value of $\psi(\theta, \theta')$. The observations on Rochelle salt, presently to be noticed, confirm this law.* If we denote by $l$ the minimum length of a double oscillation, or the space passed over during one complete period by a ray transmitted at right angles to both axes, we have $k = \frac{t}{l}$; and consequently $h = \frac{l}{t}$, $l = h t$. If we substitute for $h$ and $t$ their values above found, we obtain $$l = 0.00076497 \text{ inch}$$ for the minimum length of a period performed by a mean red ray in mica. * When $\theta = \theta'$, as in crystals with one axis, we have $\psi(\theta, \theta') = \sin \theta^2$, a result long since confirmed by the accurate experiments of Brewster and Biot. The velocity of the extraordinary ray in such crystals is given by the formula $v^2 = V^2 + \alpha \cdot \sin \theta^2$. Following this analogy, we may conclude that in crystals with two axes we should have $v^2 = V^2 + \alpha \cdot \sin \theta \cdot \sin \theta'$. Now this is precisely the expression at which M. Biot has recently arrived. This very simple and elegant result was communicated to me by that eminent philosopher in the spring of this year, and subsequently in a letter of the 2d May. His Memoir on the subject, which appears (by the Ann. de Chim.) to have been read to the Institute in April, I have not seen, nor do I know by what precise steps he was led to it, but presume it must have been by some considerations of the nature above described. In the foregoing investigation of the law of periodicity, I beg leave therefore to disclaim all intention of arrogating to myself any share in this beautiful discovery, but have thought it necessary to state the steps in the text, in order to demonstrate a truth essential to the investigations to follow, which could not have been taken for granted, or deduced by any legitimate reasoning, independent of experiment, from the equation $v^2 = V^2 + \alpha \cdot \sin \theta \cdot \sin \theta'$, by reason of our ignorance of the nature and mode of action of the polarising forces; and, have purposely abstained from entering any farther into the general laws of double refraction and polarisation than I could possibly avoid. Resuming our general equations (b) and (d) if we substitute the value now determined for \( \psi \), and write \( \frac{l'}{l} \) for \( \frac{k'}{k} \), we have \[ l' \cdot \cos \phi' \cdot \sin \theta \cdot \sin \theta' = l \cdot \cos \phi \cdot \sin (\theta - \delta a + \delta \phi). \] \[ \sin (\theta' + \delta a + \delta \phi); \] whence it is easy to derive (independent of any approximation) \[ \cos 2(a + \delta a) = \cos 2 \phi' + 2 \frac{l'}{l} \cdot \frac{\cos \phi'}{\cos \phi} \cdot \sin \theta \cdot \sin \theta'; \] while our approximate equation (d) furnishes the following very convenient formula for incidences nearly perpendicular \[ \sin \delta a = \frac{l - l'}{l} \cdot \frac{\sin \theta \cdot \sin \theta'}{\sin 2a}. \] The simplest supposition we can frame relative to the values of the constant elements \( l, l' \) is their proportionality to those of \( c, c' \), or the lengths of the fits of easy reflection and transmission. This cannot certainly be far from the truth in crystals with one axis, in which the coincidence of the tints, with those of Newton's scale, is for the most part exact. In sulphate of lime too, and mica, the only crystals with two axes which have been examined with sufficient exactness, and under the proper circumstances for ascertaining this important point, the law of proportionality seems to be sustained with great precision. This may seem to authorize the general conclusion, that in all cases, \( \frac{c}{c'} = \frac{l}{l'} \). Let us see how this agrees with the measures given in the former part of this paper. In sulphate of baryta, if we take Dr. Brewster's measure of the dispersive power,* we have \( \delta R = 0.019 \), and consequently, calculating on the data determined in page 71, we must have, at the virtual pole, \[ \phi = 21^\circ 5' 30'' \quad \phi' = 20^\circ 50' 30'' \quad \delta \phi = -15' \] Now, if we suppose \( l = 6.3463 \) \( l' = 3.9982 \), the values of \( c \) * \( \delta R = 0.019 \). Treatise on new Philosophical Instruments. and \( c' \) respectively for the extreme red and violet rays,* we shall find by substitution in our formula (h) \[ \delta a = 51' 10'' \] But a red ray penetrating the surface from within the crystal at an angle \( a = 18^\circ 12' 30'' \), and a violet one at an angle \( a + \delta a = 19^\circ 3' 40'' \), would emerge at the respective angles \( 30^\circ 59' \) and \( 32^\circ 58' 20'' \), and would include between them an angle of \( 1^\circ 59' 20'' \), which should be the apparent separation of the red and violet axes in the plate employed. Now, previous to the computation of this result, I had carefully measured this angle, by observing the incidences at which the extreme red and violet rays of the prismatic spectrum, received on the reflector of a graduated apparatus, respectively disappeared from the extraordinary image at the poles \( P, P' \). I thus found Interval of the poles \( P, P' \) for red rays - \( 62^\circ 2' \) Do. for violet - \( 66^\circ 5' \) Semi-difference, or apparent separation of the axes \( 2^\circ 1' 30'' \) which differs from its computed value only by \( 2' 10'' \). We may therefore fairly conclude, that in the case of sulphate of baryta, the hypothesis \( \frac{l}{l'} = \frac{c}{c'} \) does not deviate sensibly from the truth. If we apply our formula (i) to the measures above given for Rochelle salt, the result will be widely different. The same supposition as to the values of \( l, l' \) being made, we get \[ \delta a = 4^\circ 2' 50'' \] The incidence being nearly perpendicular, and the angle small, we need only increase it in the proportion \( 1.499 : 1 \), to have the apparent angle, which thus comes out \( 6^\circ 4' \). We have already found \( 9^\circ 46' \) for the same angle, by a method which * Biot, Traité de Physique. Vol. IV. must necessarily give a result much below the truth. This difference is by far too great to arise from any errors of observation; but to obtain more exact measures, I took several times the apparent angular separation of the axis of each colour from that of the extreme red by the direct homogeneous light of a sunbeam, separated by the prism, and received on the reflector of a divided apparatus, when, after the proper reductions for refraction and dispersion, the results were as follow: | Colour | Apparent separation of the axes | Real separation \( \delta a = \) | Values of \( 2a \) | Number of observations | |-----------------|---------------------------------|----------------------------------|-------------------|-----------------------| | Extreme Red. | 0 0 | 75 42 | | 13 | | Mean Red. | 1 33 | 73 38 | | 45 | | Do. Orange. | 2 37 | 72 14 | | 18 | | Do. Yellow. | 4 0 | 70 23 | | 20 | | Green. | 5 49 | 67 57 | | 16 | | Blue. | 8 2 | 65 0 | | 13 | | Indigo. | 10 21 | 61 54 | | 33 | | Indigo Violet. | 11 17 | 60 40 | | 2 | | Mean Violet. | 13 58 | 57 8 | | 2 | | Extreme Violet. | 15 23 | 55 14 | | 8 | Though the total separation of the red and violet axes in this table so far exceeds what we had before estimated it at, I am fully satisfied that it is no way exaggerated, but rather falls short of the truth. It is very practicable, by combinations of coloured glasses, liquids, &c. to insulate either extremity of the spectrum in a state of the most absolute purity. In this climate, the dispersed light of the sky in the neighbourhood of the sun, which always mixes with the prismatic beam, is so considerable as to obliterate the feeble rays which compose the two extremities of the spectrum, and it is only by interposing such combinations between the eye and the Iceland crystal used to analyze the polarised ray, that they can be examined with any certainty. The combination I employed for the extreme red was such, that when the whole spectrum thrown on a white screen was viewed through it, it was seen reduced to a perfectly circular, well defined, deep red image, whose centre fell on the very farthest termination of the red as seen by the naked eye, and whose circumference attained, or perhaps surpassed the point where the maximum of the calorific rays has been supposed to be situated. In like manner, when the same spectrum was examined with the violet combination, a very slightly elongated violet image became perceptible, but every trace of the indigo, and the brighter portion of the violet rays was extinguished. For observations on the indigo, and all the more refrangible portion, I employed similar artifices, without which I found it perfectly impracticable to obtain any regular and comparable results. The coefficient $\frac{l-l'}{l}$ in our formula being the only part not immediately deduced from observation, it is evident that the assumption $\frac{l}{l'} = \frac{c}{c'}$ must be widely erroneous in the present instance, and it therefore becomes necessary to ascertain the values of $l$ by direct measures. This is rendered easy by the equation (e) which gives $$l = t \cdot \frac{\sin \theta \cdot \sin \theta'}{n \cdot \cos \phi}.$$ We have only therefore to observe the inclinations of a plate of known thickness, properly cut and adjusted to 45° azimuth, which correspond to the alternate disappearances of the ordinary and extraordinary images, at which points the values of $n$ are $\frac{1}{2}, \frac{2}{3}, \frac{3}{2}, \frac{4}{3}, \&c$; computing then the values of $\theta, \theta'$, and $\phi$, and substituting, we get the values of $l$, without detailing particular experiments. The following table expresses the final result of a great number of such measures. | Colour | Values of \( l \) in inches | Number of Observations | |-----------------|-----------------------------|------------------------| | Extreme Red. | 0.0056158 | 64 | | Mean Red. | 0.0050032 | 14 | | Mean Orange. | 0.0045852 | 24 | | Mean Yellow. | 0.0040583 | 52 | | Mean Green. | 0.0036549 | 62 | | Mean Blue. | 0.0032863 | 22 | | Mean Indigo. | 0.0029868 | 52 | | Extreme Violet. | 0.0025093 | 49 | The observations from which this table was calculated, were made indiscriminately on the maxima and minima of all orders. Those of different orders were of course computed separately, and found to agree without exception in giving the same values of \( l \) within limits of error less than those to which the observations are liable; thus affording another proof of the exactness of the law of periodicity above employed. Now, if we compare these, one with another, and with those of \( c \) as deduced by M. Biot from Newton's observations, we shall have as follows: | Colour | Values of \( \frac{v}{l} \) | Values of \( \frac{c}{c} \) | |-----------------|-------------------------------|-------------------------------| | Extreme Red. | 1.00000 | 1.00000 | | Mean Red. | 0.89093 | 0.96215 | | Mean Orange. | 0.81659 | 0.9490 | | Mean Yellow. | 0.72266 | 0.85550 | | Mean Green. | 0.65082 | 0.79433 | | Mean Blue. | 0.58520 | 0.73725 | | Mean Indigo. | 0.53156 | 0.69641 | | Extreme Violet. | 0.44684 | 0.63000 | It appears from this comparative statement, that the forces of polarisation and double refraction in the body now examined, act with much greater proportional energy on the more refrangible rays than in mica, sulphate of lime, and MDCCCXX. other similar bodies, and consequently that, even were its axes coincident, its tints, though perfectly regular, would still differ very sensibly from the colours of thin plates. This secondary cause of deviation ought to become sensible in plates cut so as to contain both axes, if examined at a perpendicular incidence; but I have not yet had an opportunity of making the trial. If we calculate on the numbers above given, it will soon appear that a perfect coincidence of all the colours in a single virtual pole is impossible. For this purpose we may employ our equation (i) which easily affords the following \[ \cos 2(a + \theta) = \cos 2a \left\{ 1 + \frac{2l}{l-l'} \cdot \tan 2a \cdot \sin (-\delta a) \right\} \] taking \(M\) an auxiliary angle such that \[ \tan M = \sqrt{2} \cdot \tan 2a \cdot \frac{l}{l-l'} \cdot \sin (-\delta a) \] whence the value of \(\theta\) or the position of the coincidence of any two coloured rays becomes known, the values of \(l, l'\), and \(-\delta a\) being given from the foregoing tables. If we unite the mean red with the mean green, these formulae give \(\theta = -11^\circ 29'\), and if with the mean blue, \(\theta = -14^\circ 8'\), of which the one falls short of, and the other exceeds the angle \(-13^\circ 1'\) given by observation. If we determine by interpolation the values of \(l'\) and \(-\delta a\), which give \(\theta = -13^\circ 1'\), we shall find very nearly \[ l' = 34581 \quad -\delta a = 3^\circ 37' \quad -\delta a + 1^\circ 2' = 4^\circ 39' \] which correspond to a blue ray strongly inclining to green, and in the brightest part of the colour. Now it is evident that when a rigorous union of all the rays in the proportion in which they exist in white light, is impossible, that of the strongest and brightest colours in opposition to each other will at least ensure the nearest approach to a virtual pole on the principles above demonstrated, and a white will thus be produced, not indeed mathematically perfect, but containing no marked excess of any of the more powerful colours. The apophyllite is the only crystal with one axis whose tints exhibit a sensible deviation from the scale of Newton. Its phenomena, however, are entirely independent of the first and principal cause which produces the deviation in crystals with two axes, viz. the separation of the axes of differently coloured rays, and are referable solely to the secondary and subordinate cause, of which Rochelle salt has just afforded an example, viz. a peculiarity in the law which regulates the lengths of the minimum oscillations of the differently coloured rays within the medium. 1. The tints of the apophyllite commence at the centre of the rings and increase in regular progression outwards, following the same order, whatever be the thickness of the plate. It follows from this, that the multiplier $M$ in our general formula, (a) is the same for all the coloured rays, being zero at the commencement of the scale; and hence it follows, as a necessary consequence, that the axes of all the colours are united in one, and the virtual and actual poles coincide with each other and with the centre. Did any sensible separation of the axes exist, it must become perceptible by the ellipticity of the rings when examined with homogeneous light of that colour from which they are farthest asunder; but with the greatest attention, in plates of considerable thickness, I have not been able to observe the slightest shifting of the axis, or deviation from the circular figure, in passing from a red to a Moreover, it is evident from the preceding theory, that any difference which may exist in their position, if too small to be sensible to the eye, can produce only an imperceptible deviation of tints. In fact, if we suppose \(a = 0\) for any colour, we get, for the position of the virtual pole, \[ \sin \theta = \sqrt{\frac{1 - \nu}{l}} \cdot \sin \delta a \] \( \theta \) being the angular distance of the point of coincidence from the single axis of that colour. It is, consequently, insensible when \( \delta a \) is so. Now, the polarising force of the apophyllite being very feeble, the diameters of the rings in any plate of moderate thickness must so far exceed this very minute quantity, that the virtual poles, did any exist, must fall within the limit of the central blackness; the Newtonian scale would still appear to commence from the centre, nor could any sensible deviation from it arise from this cause. 2. When the prismatic spectrum is passed over an apparatus containing a plate of this mineral, no perceptible change in the magnitudes of the rings for different colours takes place. Hence it appears that the value of the function \(l\) for all the coloured rays is nearly alike. By measures taken on a divided apparatus, a slight difference is observed. Taking the mean refractive index \(R\) at 1.5481 (by a very careful measure) and the dispersion \( \delta R \) at 0.017, the formula \[ l = t \cdot \frac{\sin \theta^2}{n \cdot \cos \phi} = \frac{t}{n} \cdot \sin \theta \cdot \tan \theta \] gave as follows: | Extreme Red. | Mean Red. | Mean Orange. | Mean Yellow. | Mean Green. | Mean Blue. | Mean Indigo. | Extreme Violet. | |--------------|-----------|--------------|--------------|-------------|------------|--------------|----------------| | \(l = 0.0093066\) | \(0.0092810\) | \(0.0092337\) | \(0.0091503\) | \(0.0090643\) | \(0.0092059\) | \(0.0093904\) | \(0.0100660\) | This table, though not given as exact, owing to imperfections in the specimen examined, agrees with the succession of tints which, as far as the fourth order, were as follows: **Apophyllite. Thickness = 0.0829.** | Incidence | Ordinary pencil | Extraordinary | |-----------|-----------------|---------------| | 0 | Bright white | Black | | 13 | White with a trace of purple | White slightly greenish | | 21 | Exceeding sombre violet | Pure bright white | | 25 | Pale greenish yellow | Purplish white | | 29 | White | Sombre violet blue | | 30 | White | Extremely sombre violet | | 33 | White with a strong tinge of violet | Pale yellow green | | 35 | Blue strongly inclining to purple | Greenish white | | 37 | Sombre indigo inclining to violet | White | | 38 | Sombre violet | White | | 40 | Tolerably good yellow green | Purplish white | | 43 | White with a trace of yellow white | Obscure indigo inclining to purple | | 44 | Pale purple | Sombre violet | | 46 | Sombre purple blue | Tolerable yellow green | | 49 | Sombre violet | Yellowish white | | 50 | Green yellow | White | | 53 | Yellowish white | Pale purple | | 56 | Yellowish white | Sombre indigo blue | | 59 | Pale purple | Sombre violet | | 61 | Sombre indigo | Livid grey | | 62 | Sombre violet | Yellow green | | 63 | Faint violet white | Pale yellowish white | | 66 | Livid grey | Yellowish white | | 69 | Tolerable green yellow | White | | 70 | Yellow white | Purple | In the colours of thin plates and others of the like composition, the difference in the lengths of the periods of the different rays is so considerable, that after seven or eight alternations of colour the rings confound one another, and are blended into a uniform whiteness. Were the periods more nearly equal, a greater number of rings should be visible, and were they strictly so, the succession of alternate whiteness and blackness should be continued to infinity. As the values of in the apophyllite approximate pretty closely to this limit, we should expect to see a much greater number of rings, and this I find to be really the case. By enclosing a thick plate in balsam of copaiba in a proper apparatus to increase the range of incidence, I have counted as far as the 35th order, when I desisted; not from any want of alternate colours, but owing to their extreme closeness, which rendered it impossible to number them distinctly. Indeed I have no doubt, that could a very thick and limpid specimen be procured, hundreds might be seen without artificial aid. In two instances then, at least, and probably in many more, or perhaps to a certain small degree in all cases, the minimum lengths of the periods deviate in their respective proportions from those of the fits of easy transmission and reflection; a circumstance which of itself is sufficient to prove the independence of the causes of these laws of periodicity. If we take $R r = RA$, Fig. 5, Pl. V. and construct a curve whose abscissas $AP$ are the values of $c, c', \&c.$ and ordinates those of $l, l', \&c.$ the straight line $roygbiv$ inclined at $45^\circ$ to $AR$ will represent the locus for crystals, such as carbonate of lime, in which the periods follow the Newtonian law, $r'o'y'g'b'i'v'$ will represent the same locus for tartrate of soda and potash, while $r''o''y''g''b''i''v''$ is the curve similarly traced for apophyllite.* * Having communicated to Dr. Brewster my observations on the deviation of tints, and the conclusion I had thence deduced as to the separation of the axes of the differently coloured rays, I received in answer a letter, from which, in justice to that indefatigable observer, I subjoin the following extracts. "My Dear Sir, Esk Hill, by Roslin Laswade, Sept. 18, 1819. "In consequence of having been some time from home, I have only now received your letter, and hasten to reply to that part of it in which you request me to state what results I had obtained respecting the deviation of the tints from Newton's scale. The following general points will enable you to judge of the progress which I had made in the enquiry. "1. In almost all crystals with two axes there is a deviation from the tints of Newton's scale. "2. This deviation is greater in some crystals than in others, being a maximum in acetate of lead and tartrate of potash and soda. "3. That all these crystals may be divided into two classes, viz. those which have the red ends of the rings inwards and the blue ends outwards, and those which have the red ends outwards and the blue ends inwards. "4. That in all crystals with two axes, the doubly refracting force of one axis in general acts differently upon the differently coloured rays from the doubly refracting force of the second axis. "5. That as the polarising force is always proportional to the force of double refraction, the polarising force of one axis will act differently on the differently coloured rays from the polarising force of the other axis. "6. "7. The consequence of this is, that there will be different resultant axes, or different points of compensation for the differently coloured rays. "8. All these effects may be calculated with the utmost accuracy, if the ratio of the dispersive powers of the two extraordinary refractive forces is given, or vice versa, the dispersive powers may be obtained from the angles of the resultant axes for the red and violet rays of the spectrum. "9. I have found crystals in which these phenomena are decidedly connected with the rotatory phenomena; and from this highly important fact I am led to conclude, that both have the same origin, and that all the rotatory phenomena are, as I have stated in my paper, the result of the uncompensated tints of two axes, equal for the mean ray, but unequal for all the rest. (Here follows an illustration by a diaphragm.) "10. The division into two classes in § iii. as founded merely on observation, is converted into another division into two classes, viz. 1. That in which the doubly refracting force of the principal axis acts more powerfully on the blue rays than the other axis does; and 2. That in which it acts less powerfully. The first class comprehends those crystals in which the blue ends are inwards, and the second those in which the red ends are inwards, or nearer the principal axis." In a subsequent letter (Oct. 4), he adds, "The virtual poles, which you mention, I discovered in the year 1815, and I have two accounts of them in my Journal, the one signed on the 24th January, 1816, and No comments on the above extracts are necessary. They establish at once the priority of Dr. Brewster's observations, and the independence of mine. With regard to the division of crystals into two classes, which observation has alike suggested to both of us, it is unnecessary, if we regard either of the two classes as having the angle between the resultant axes greater than a right angle. In Dr. Brewster's table, Phil. Trans. 1818, p. 230, succinic acid and sulphate of iron are stated as having this angle $90^\circ$. If this determination corresponds, as in all probability it does, to the yellow rays, they belong at once to both classes, and are, in fact, instances of the limit where one class passes into the other. Bi-carbonate of ammonia, in which I can perceive no separation of the axes of different colours, nor of course, any virtual poles, belongs in like manner to both classes, or to neither. JOHN F. W. HERSCHEL. APPENDIX. Description of an instrument employed in the foregoing experiments on the polarised rings. The singular property possessed by the tourmaline, by which a plate of it of any moderate thickness cut in a direction parallel to its axis of double refraction, is enabled to absorb the whole, or nearly the whole, of an incident pencil polarised in a plane parallel to that axis,* was pointed out by * The same property is observable in the epidote, the axinite, and all other natural and artificial crystals which exhibit any degree of dichroism when examined by unpolarised light. Muriate of palladium and potash possesses it in the highest perfection. This remarkable effect is easily explained by a reference to the general principles laid down by Dr. Brewster in his paper on absorption, Phil. Trans. 1819, p. 11. The incident pencil is separated by the doubly refractive force into two, oppositely polarised, one of which is partly absorbed, the other emerges (polarised in its proper plane) of nearly its original intensity. M. Biot, in the fourth vol. of his Traité de Physique, and he has availed himself of it with his accustomed ingenuity, as affording an extremely ready and convenient mode of viewing the phenomena of polarisation, much more so than by the use of plates of agate, prisms of Iceland spar, or a second reflection. It follows, from the above mentioned property, that if a beam of ordinary light be made to traverse such a plate, the whole of the emergent pencil, or nearly so, will be polarised in a plane at right angles to the axis; for the incident ray being divided by the doubly refracting force into two pencils, polarised in planes, the one parallel, the other perpendicular to the axis, the former is extinguished in its passage, while the latter emerges with nearly its full intensity. Hence, if two such plates are crossed at right angles, though separately very transparent, their combination will be opaque. There is a great difference, however, in the degree in which tourmalines of different colours possess this power. Those of a light green, pink, or bluish colour, are quite improper, allowing a considerable portion of light to pass when so crossed, while, on the other hand, those whose colour verges strongly to the honey yellow, or to the hair brown, or purplish brown, effect nearly a complete absorption, and afford, when crossed, a combination almost impervious to light. In ignorance of this distinction, I sacrificed several fine and valuable specimens before I could obtain proper plates. When a crystallized lamina, cut in a proper direction, is interposed between such a combination of plates, it disturbs the polarisation which the light has received in traversing the first plate, and renders a certain portion of it capable of traversing the second: the colour and intensity of this portion varying with the direction of the ray, give rise to the phenomena of the polarised rings, which may accordingly be seen by applying the eye, and receiving on it the dispersed light of the clouds, &c. In order, however, to equalize as well as disperse the light, which is of great importance to obtaining a perfect view of the phenomena, an emeried glass may be cemented on the anterior plate, or the first surface of the plate itself roughened; but it will be found more convenient in practice to employ a double convex lens of short focus for this purpose, by which, if necessary, a very strong illumination may be obtained, and an extremely minute portion of a crystal subjected to examination. I have thus, occasionally, examined the rings in a portion not exceeding the hundredth of an inch in diameter, and thus detected irregularities of crystallization of a very singular nature, in many bodies, which would have eluded any other mode of observation. For this purpose, the crystal must be cemented over a small aperture in a thin sheet of brass, on which the focus of the lens must be exactly adjusted to fall.* If, instead of applying the eye to receive the light so dispersed, we place a screen at some distance in a darkened room, the apparatus is converted into a solar microscope, and the rings will be seen projected on the screen. The con- * I have now an apparatus preparing, in which the first plate of tourmaline itself is formed into a double convex lens, by which the loss of light at two surfaces will be suppressed. It is easy to adapt such a lens to a double microscope, for the purpose of detecting microscopic irregularities; and I have reason to suppose a variety of curious results will be brought to light by these means. crystallized bodies on homogeneous light. struction of the apparatus I employed is as follows. AB is a brass tube, within which are fitted, 1st, a fixed diaphragm, \(aabb\), carrying the first plate of tourmaline in its centre; 2dly, a diaphragm, \(ccdd\), moveable freely in its own plane by means of the pin \(g\) passing through a slit in the side of the cylinder A B, which occupies an arc of about 120° of its circumference. This is destined to receive the crystallized plate \(dd\), while a cylinder, \(hhfff\), made to slide and turn smoothly within A B, carries the second tourmaline, \(ff\). It is essential that the tourmalines employed for this purpose, and especially the posterior one, should be perfectly free from all flaws and blemishes; but large plates not being required, this condition is easily satisfied. The plates so arranged, and brought as near together as possible, the extremity A, of the cylinder A B, is fitted to slide somewhat stiffly on the brass tube P Q, furnished with a lens L, of about two inches focus, and a screw PP, by which it can be adapted to the apparatus usually employed for reflecting a sunbeam into a darkened chamber. The sliding motion of the cylinder A B allows the focus of the lens to be adjusted so as to fall exactly on the first surface of the posterior tourmaline \(f\), while its rotation suffers the axis of the anterior one to be placed perpendicular to the plane of reflection. By this arrangement two advantages are gained. The reflector employed (though metallic) always polarises a more or less considerable portion of the reflected beam, which in any other position is partially, or totally extinguished by the first tourmaline, and a great loss of light ensues, which it is of the utmost consequence to avoid: moreover, by this disposition, the action of the reflector is brought to conspire with that of the tourmaline, and the polarisation of the light which traverses it (which is never rigorously exact) is thereby rendered more complete. It is convenient to have sliding tubes containing lenses of different focal lengths according to the crystal examined, for the intensity of illumination of any point in the screen being, *caeteris paribus*, as the square of the focal length, consequently, when the rings lie within a very small angular compass, a greater illumination of every part of them may be obtained by using a lens of a longer focus. The dimensions of the figure, Fig. 6. Pl. V. are nearly of the actual size.