so that the prismatic apparatus may be moved at pleasure through the whole focal distance AF. We begin by determining its posi tion in the focus. For this purpose, we direct the telescope to the object, and move the prisms towards the eye, until the two images formed at the focus exactly coincide. We then read the point of lateral division, to which the index, carried along with the prismatic apparatus, corresponds; this point is the zero, from which the distances Fc are to be counted. Suppose, for instance, that it answers to the number 50. When we have observed any object and brought the images into contact, we read anew the point of division to which the index answers. Suppose this number to be 125; then, for this value of the apparent diameter, Fc will evidently be equal to 125 - 50, or 75 parts of the division. In order to know, once for all, the ratio of this number to the apparent diameter, we employ a circular Fig. 96. or spherical object SS' of a known diameter, placed at a distance AC, ascertained by direct measurement, or by a trigonometrical operation. We may hence calculate the apparent diameter SAS', subtended by this object, seen at the distance AS. This being done, we observe the same object through the prismatic telescope, placing the object-glass at the same point A; and when we have brought the two images into contact by the motion of the prisms, we measure on the lateral division the distance Fc. Then the ratio of this distance to the apparent diameter SAS' is known for all cases, and we can employ it in all our other observations. Or rather we may make use of it in calculating beforehand the apparent diameters which answer to a certain number of values of Fc, and write them on the tube against each distance. This is commonly done in instruments intended for use. But instead of expressing the apparent diameters in minutes and seconds, we express the ratio between the distance of the object and its magnitude, which enables us to deduce one of these elements from the other. Thus, from the mean height of the men who compose a body of troops, we can determine their distance; we can likewise determine the distance of a ship at sea, from the supposed height of its mast. Our results, however, are the more liable to error, according as the distance is greater and the object smaller; so that we must not think of employing this expedient for the purpose of ascertaining the distance of the heavenly bodies, although it has been sometimes proposed. Their apparent diameter is the only element to be determined by means of it.

136. Hitherto we have supposed the first edge F of the ordinary image FF', to be precisely in the axis of the object-glass at the moment when we observe the contact. This condition is absolutely necessary in order that the incident ray Al, which, after its division, comprehends the ordinary image, may traverse the prismatic apparatus perpendicularly to its exterior surfaces, which is the only case we have yet considered. But if the observed object were a heavenly body, whose motion caused it to pass successively through the whole field of view of the telescope, what would be the result? Mathematically speaking, the value of the angle Fcf, would no longer be constant in the different stages of its passage. If these variations are insensible, which is the case when the refracting angles of the prisms are very small, we can establish the contact of the two images when the heavenly body has entered the field of view, and it will continue through the whole extent of this field; but if we much enlarge the opening of the angles of the prisms, and the deviation which is consequent upon it, the angle Fcf will begin to vary sensibly for the different incidences which the field of view admits of, and the images after they are once brought into contact, will separate in traversing it. To avoid this inconvenience, Rochon proposed to substitute in the place of double prisms of a great angle an assemblage of several similar prisms, but each of a very small angle, and cemented together in such a manner that ail the principal sections should coincide exactly in the same direction. Indeed, in such a system, the separation of the rays increases with the number of double prisms, and the influence of the variation of incidences on the divergence of the images is much less sensible than in one double prism only, which gives an equal divergence; for this, we might easily give a reason founded in theory. But great care must be taken that the superposition of the prisms be exactly according to the principal sections, in order that the images may not be multiplied beyond two; and we must also use certain precautions in cutting the prisms, that they may not be coloured. These particulars, taken together, render the process in question, almost impracticable, at least for astronomical purposes.

137. In what precedes, we have supposed the images FF', ff', formed by the object glass at the focus, to be observed by the naked eye; whereas, in general, we look at these images through a magnifying glass, or a system of glasses disposed in

such a manner as to enlarge them indefinitely without rendering them indistinct. This is called the eye-glass, because it is situated in the part of the telescope next the eye, in the same manner as the former is called the object-glass, because it is situated towards the object. But as the action of the eyeglass is subsequent to the formation of the double images, it can evidently have no influence on the existence or non-existence of their contact, but only enables us to judge of it with greater precision. Thus all we have said on the supposition that we use the naked eye, is equally true in the case where the eye is provided with a glass; accordingly we have not had reference to this modification, in the account we have given.

The celebrity of Rochon's micrometer, and the numerous applications which are made of the process of division of images, on which the instrument depends, have rendered it necessary to go into this minute description of it. Still it is not exempt from inconvenience in practice, especially when we would apply it to cases of great magnifying power, as is required in astronomical observations. Accordingly, M. Arago has devised another and a decidedly preferable way of employing double refraction, which will be made known hereafter.

Of certain Singular Appearances occasioned by Double Refraction..

137. WHEN We look at small objects through a rhomboid of Iceland spar, the disposition of the two images presents certain peculiarities, which are so many consequences of the theory. These did not escape the notice of Huygens. As they might embarrass persons unaccustomed to considerations of this kind, it may be worth while to take a brief notice of them.

Let L be the radiating point, and S the eye, which, for greater Fig. 97. simplicity, we will suppose situated in the plane of the principal section ABA'B' of the rhomboid. Let us inquire in what manner vision will take place under these circumstances.

Here, as in the case of refraction through any other transparent body with parallel faces, there will be an ordinary ray which can reach the eye. Let LII' be this ray. As it enters Opt.

18

1

at I, by the first face of the rhomboid, there will be an extraordinary ray II; but this cannot come to the eye. To be assured of this, we have only to draw through the point of incidence I, the line IA, parallel to the axis of the rhomboid. The force emanating from this axis will repel. the luminous particles, and those which yield to it, must evidently be removed beyond the ordinary ray, taking, for instance, the course II. But having arrived at I, in the second surface, they will emerge parallel to their primitive direction LI. Hence there results a ray IS', which being parallel to I'S, cannot pass through S, where the eye is placed. For a still stronger reason, the same may be said of every other extraordinary ray proceeding from the incident rays, which approach nearer to the obtuse angle A'. The extraordinary image will therefore be given by the incident rays which diverge from LI in the contrary direction, that is, which approach the solid angle B'. Among these, there will be found one, as Li, whose ordinary branch cannot come to the eye, but whose extraordinary branch ii, may come to it after its emergence; so that the eye will receive two images, the one ordinary in the direction SI', the other extraordinary in the direction Si; and the first will always appear nearer the small solid angle B', than the other. If the position of the eye and that of the radiating point, with respect to the rhomboid, be given, we can easily calculate the directions of these two rays, by taking as unknown quantities the angles of incidence and emergence, which they form with the two faces of entrance and emergence. Because, for each ray, ordinary or extrordinary, these angles must be equal to each other. But the result being one of mere curiosity, it is sufficient to have indicated the general course. We can prove the crossing of these rays in the interior of the rhomboid, by a very simple expedient devised by Monge. Things being disposed as in the figure, we pass a card slowly over the surface A'B', situated towards the radiating point. When it has come to i, it will intercept the incident ray Li, which gives the extraordinary image. We shall then see the emergent ray Si disappear, although from the direction in which it seems to come, we should expect to see the ray SI' disappear first. In order that this priority may be accurately observed, we must place the eye very near the rhomboid, which will enlarge the angle S; and in order to have a luminous object of a small diameter,

we may look at the light of the sky, through a small hole pierced in a card, or what is still better, at a black dot made on a white sheet of paper. In each case, the luminous object must be placed at a considerable distance. For the more the radiating point L, approaches the surface of the rhomboid, the more will the point K, where the rays cross each other, approach also this surface. But however near we suppose L, provided it be without the crystal, so that the formulas of Huygens may be applied to the luminous rays which emanate from it, the intersection of the two rays will take place in the interior of the rhomboid; and consequently, the phenomenon we have described, will continue to occur, although with different divergences. Figure 98 represents the case where the radiating point and the eye are situated in the same straight line perpendicular to the faces of the rhomboid.

138. When we look thus, through a rhomboid, at the two images of a luminous point, whatever position we otherwise give it, the ordinary image will always appear nearer the eye than the extraordinary image; this likewise is a consequence of the theory.

In order to understand the reason of it, let L represent the radiating point, S the centre of the eye, and SI' the ordinary ray refracted in passing through the medium ABA'B', whose opposite faces are parallel. If the eye were only a mathematical point, the ray I'S, would be the only one of its kind which could come to it; but as the pupil has a certain extent, it will be seen that it must receive also a certain number of other ordinary rays near to SI', and consequently derived from incident rays. near to LI. These rays departing from L, form a cone which has its vertex in L; they form another cone when refracted by the medium, and another still when they emerge from the second face AB. Now if we calculate the distance SL", from the eye to the vertex of this last cone, whatever be the nature of the intervening medium, whether it be crystallized or not, we shall find that the point L" is always nearer the eye, than the point L; and the more so, according as the medium has a greater refracting power; because its refraction increases the divergence of the emergent rays. Now it is precisely this divergence, which enables us, in general, to judge of the distance of the luminous points and their images; so that the refracted image

Fig. 99.