which he named a refractometer. His paper was subsequently published, with a plate, in the Monthly Microscopical Journal." The principle made use of in applying this instrument was the increase in the focal length of the object-glass of a microscope, caused by looking through media of different refracting power. The author showed that if t be the thickness of this medium, and d the amount of the displacement of the focus, the index of refraction μ may easily be calculated from the following equation:

In the instrument described by Dr. Royston-Pigott, the amount of this displacement, and also the thickness of the object under examination, were determined by means of a micrometer screw fixed under the stage of the instrument, in such a manner that it became unsuitable for use as an ordinary microscope.

At the time of the reading of this paper I was much struck with the general method employed, and in the subsequent discussion I said that probably some modification of it might prove very useful in studying minerals. I have now succeeded in proving this very_completely.

From the first I was anxious to contrive some arrangement that would enable us to obtain the necessary data with an ordinary microscope, or at all events with one so slightly modified as not in any way to interfere with its general use; and I think that I have succeeded in accomplishing this by a very simple addition, which will also enable us to use the instrument for a number of purposes not originally contemplated.

Practically, the application of the method I propose is very simple. If an object be placed on the stage of a microscope and the focus adjusted, on placing over it a plate of some highly refracting substance the focal length is increased, and hence, to bring the original object into focus, the body of the microscope must be moved farther from it. In order to measure the amount of this displacement, nothing, therefore, is required but some means for accurately measuring the distance over which the body of the microscope is thus moved. This may be roughly done with a small scale, accurately divided to oths of an inch; but it is far better to have an attached scale and vernier, so as to be able to read to Tooo of an inch, and to estimate half that quantity. The thickness of the specimen is easily measured by focussing first the particles of dust on the surface of the glass plate supporting the mineral, and then those on its upper surface. Several observations should be made of the position of these different planes, as shown by the readings on the scale, and the means taken, in order to

* Vol. xvi., 1876, p. 294.

compensate for small accidental errors, and care must of course be taken to avoid any that might be caused by imperfections in the instrument. If the section of the mineral be covered with thin glass, which in most cases is very desirable, its apparent thickness must be measured, and due allowance made for it in calculating out the results. It is also requisite to deduct from the indices given a small quantity due to the effects of

t

t d

by the formula μ = spherical aberration which varies with the aperture and correction of the object-glass, and also with the value of the index in each particular case. In order to obtain as accurate measurement as possible, a number of precautions must be taken, which are all simple enough, but it would occupy too much time to describe them in detail. With proper care the errors in the values of t and d ought to be certainly less than rooo of an inch. The accuracy with which the indices of refraction can thus be determined depends much on the thickness of each specimen, but if it be from

to of an inch, the errors ought to be limited to the third place of decimals. In practically employing this method it is of great importance to have some object which gives a very definite focus. In the first instance I made use of a glass plate having very fine parallel scratches, made with the finest emery paper; but I soon found that it would be very convenient to have more definite and equidistant parallel lines, not in any way affected by moving the stage. This can be accomplished by having them ruled of an inch apart on a glass plate, fixed as far as possible below the lenses of an achromatic condenser, with a small central stop, which gives at the focus a much reduced image easily adjusted either a little below the lower or upper surface, or nearer the centre of the specimen, according as its shape may make it necessary, so that the light may pass to the object-glass as equally as possible from all sides. It is also extremely useful to have an iris diaphragm fixed just below the grating, so as to be able to obtain an image of a circular hole of any requisite diameter. I had two sets of lines ruled on the same surface at right angles to each other, in order that there might be less chance of mistaking any striæ in the mineral for a single system of lines, and that either system might be used if the other were obscured. This arrangement has fortunately led to the discovery of an entirely new class of optical properties.

Unifocal and Bifocal Images.

On looking at the double system of lines without any intervening object, both sets of parallel lines are seen at the same focus. If a plate, with parallel flat surfaces of glass or of any transparent

mineral which has no double refraction, be placed on the stage of the microscope, with its surfaces perpendicular to the line of vision, the two systems of lines can still be seen at the same focus, no matter what may be the azimuth of the lines to the axes of the crystal. The image may thus be said to be unifocal, and to have no special focal axis. The index of refraction, determined as above explained, is that of an ordinary ray. On the contrary, if the mineral possesses double refraction, the phenomena seen by means of the extraordinary ray may be totally different, and as though a cylindrical lens had been placed in front of the object-glass. In order to be able to examine separately the two rays polarized in opposite planes, a Nicol's prism must be used over the eye-piece, arranged at such an azimuth as to transmit one or other ray alone. The ordinary ray has just the same properties, and is strictly unifocal, no matter what may be the direction of the section of the crystal; but the characters of the extraordinary ray differ greatly, according as the section is cut perpendicular, oblique, or parallel to the principal axis. I cannot refer to a better example than calcite. On examining the image of the circular hole and of the grating through a section parallel to the axis, the plane of polarization of the Nicol being arranged perpendicular to the axis of the crystal, so that only the extraordinary ray is transmitted, it will be found that at two different foci the circular hole is elongated in opposite planes, and that both sets of lines are invisible, unless they are nearly parallel and perpendicular to the axis, and that there are two focal points, separated from one another by an interval somewhat more than one-eighth of the thickness of the section, at each of which only one system can be seen at once. The image is thus truly bifocal, and has a definite focal axis, and the lines are distinctly visible only when parallel or perpendicular to this axis. When determined in the manner already explained, the index of refraction for the lines parallel to the principal axis of the crystal is the true index of the extraordinary ray, whereas that for the lines perpendicular to this axis is only an apparent index, and is equal to the square of the index of the ordinary ray, divided by that of the extraordinary.

The striking difference between a unifocal and a bifocal image becomes at once intelligible if, instead of a grating, we examine through the mineral the image of a small circular hole, as Fig. 1. In the unifocal image this is seen undistorted, well defined all round at one definite focus; whereas in the bifocal image there is no focal point whatever at which the hole can be seen of its true size and shape. There is one focal point for the two opposite sides of its circumference which are parallel to the focal axis, and at this focus the circle is drawn out parallel to that axis into a long band, and there is another focal point for those parts of the circum

ference which are perpendicular to the axis, and the image is then drawn out in a direction perpendicular to that of the former image, as shown by Figs. 2a and 26. At an intermediate focal adjustment we see merely a large circle without any definition. It therefore follows that the series of black points forming a line would be similarly drawn out at the two foci into lines, and if these overlapped, as they would if the line were at that particular azimuth, we should appear to have a well-defined black line, whereas at other azimuths this line would be spread out into a band, and so diluted with white light, as to be practically invisible. In a section parallel to the axis the images of the small hole are directly superimposed, but if we examine it through a section parallel to the cleavage they are widely separated in the plane of the principal axis, as shown by Fig. 4, and appear to lie at different levels. That due to the ordinary ray remains in the centre of the field, and is not in any way distorted, whereas that due to the extraordinary ray is thrown out of the centre from the line of axis, and is both distorted and fringed with colour. This image is very decidedly bifocal, but one system of lines is much obscured by coloured fringes, unless we illuminate with the approximately monochromatic light transmitted by red glass. When the section is cut in planes more and more. inclined to the axis, the bifocal image becomes more and more nearly unifocal, and when the section is perpendicular to the axis it is unifocal, but can be distinguished from that due to the ordinary ray by causing the light to pass obliquely. We then see two images with both sets of lines, at perfect focus, directly superimposed at two very widely separated levels, as though there were two sets of lines ruled on opposite sides of a glass plate. One gives the true index of refraction of the ordinary ray, and the other an apparent index, which is equal to the square of the true index of the extraordinary ray, divided by the true index of the ordinary. On examining the small circular hole it is seen undistorted, in perfect focus, at two widely separated foci, surrounded with a large nebulous circle, due to the other image seen out of focus, as shown by Fig. 3.

All these phenomena are totally unlike what can be seen with the naked eye in looking directly through sections cut either parallel or perpendicular to the axis. A white or black spot placed close to the specimen is then not even divided into two. The phenomena seen with the microscope depend entirely on the power of the object-glass to collect divergent rays. In the case of substances having no double refraction, this divergence merely obeys the laws of ordinary refraction, and enables us to measure the index in the manner already explained; but, in the case of the extraordinary ray, the light is bent from the normal line unequally and in opposite directions, and may thus enter the object-glass at

an angle of divergence greater or less than that depending on the index of refraction, according to the direction of the section, and to whether the double refraction is negative or positive. Thus, for example, in the case of calcite cut perpendicular to the axis, the light diverges equally all around the axis, less than normally, and therefore the focal point of objects seen through the section is made abnormally short, and the apparent index abnormally small, being, in fact, only 1.332, whereas the true index of the extraordinary ray is 1.480, and of the ordinary 1.658.

Crystals like orpiment or aragonite, which have two optic axes and three different indices, have no ordinary ray, and no permanently unifocal image, but two bifocal images polarized in opposite planes. We may thus have four different apparent indices. In the case of orpiment the image of a small circular hole is drawn out at two different foci into two crosses, as shown by Fig. 5. Each cross is produced by the combination of two bands of light polarized in opposite planes, each due to an extraordinary ray, analogous to the single extraordinary ray of calcite. In the case of aragonite, cut perpendicular to the principal axis, the arms of the crosses are nearly equal, but spread out in the manner shown by Fig. 6. This spreading out varies according as the aperture of the objectglass is large or very small. If the section is in a plane somewhat oblique to the principal axis, one bar of the cross is distorted into an irregular circle, and one arm of the other bar is spread out into a sort of crescent.

A remarkable peculiarity of crystals which thus give two wellpronounced bifocal images, is that though they may be perfectly transparent, and distant objects distinctly visible through them with the naked eye, the systems of lines at right angles to each other are perfectly invisible with the microscope, except at particular azimuths. I was extremely surprised at this fact when first I observed it, and could not understand the reason of this apparently strange peculiarity.

When the section is cut parallel to the principal and to one of the secondary axes, we obtain a cross with unequal arms at four different foci; and when cut parallel to the principal, and along the diagonal of the secondary axes, one image has the bifocal character very strongly developed, and the other is almost or quite unifocal, but can be shown to be also due to an extraordinary ray, by causing the light to pass obliquely.

If we wish to ascertain the real value of the indices, we must bear in mind the following facts: The image due to the light passing through substances not possessing double refraction, or to the ordinary ray of crystals belonging to the rhombohedral and dimetric systems, has no special focal axes, and the apparent index is the true index, no matter what may be the direction of the