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IV.—On the Question of a Theoretical Limit to the Apertures of Microscopic Objectives.

By Professor G. G. STOKES, M.A., D.C.L., LL.D., Sec. R.S., Lucasian Professor of Mathematics in the University of Cambridge.

(Read before the ROYAL MICROSCOPICAL SOCIETY, June 5, 1878.)

I HAVE just received from Mr. Mayall, jun., a photograph of Professor R. Keith's computations relative to an immersion microscopic objective by Mr. Tolles. I have not at present leisure to go through this long piece of calculation, which I am the less disposed to do as the calculation is perfectly straightforward, and has evidently been made with great care, and I can see no reason to question the result. The only reason for scepticism as to the results of such calculations seems to be a notion derived from a priori considerations, that it is impossible to collect into a focus a pencil of rays emanating from a radiant immersed in water or balsam of wider aperture than that which in such a medium corresponds to 180° in air, or, in other words, than 2 y, where y is the critical angle.

I do not wish to enter into controversy on the subject, or to criticise the arguments by which this statement has been sustained; I prefer to show directly that it has no foundation.

To disprove an alleged proposition, the shortest and least invidious plan is often to show by one or more particular instances that it is untrue.

Suppose a pencil of parallel rays is incident upon a refracting medium of index μ, and let it be required that it be brought without aberration to a focus q within the medium. By a wellknown proposition, the form of the surface must be that of a prolate ellipsoid of revolution generated by the revolution of an ellipse of which q is the further focus, and μ the eccentricity, about its major axis, which is parallel to the incident rays. Conversely, if q be a radiant within the medium, the emergent rays are parallel to the axis.

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The limit of the incident parallel rays in any section through the axis is the pair that touch at the extremities of the minor axis. Consequently in the reversed pencil the limiting rays are those that proceed from q to the extremities of the minor axis. If we suppose the index to be 1.525, for which y = 40° 59′, the extreme rays will be inclined to the axis at the complementary angle 49° 1'. Hence a radiant within glass may send a pencil of aperture 98° 2′, which by a single refraction shall be brought accurately to a second focus at infinity. The double of the critical angle is only 81° 58', so that the aperture exceeds that supposed limit by 16° 4.

If it were required that the pencil after the single refraction. should converge to a real focus, the surface would have to be generated by the revolution of a cartesian oval instead of an ellipse. If

the distance of the point of convergence were considerable compared with the dimensions of the glass, it is evident that the oval would not differ much from the ellipse considered in the first instance, nor would the extreme aperture in glass fall much short of the limit assigned above. Or again, the rays emerging from the ellipsoid might be brought to converge to a second focus q' in air by receiving them on a prolate spheroid of which q' is the further focus and the eccentricity, and allowing them to emerge from the glass by a spherical surface of which q is the centre. Or the parallel rays might be brought to a focus without sensible aberration as is done in telescopes.

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I do not, of course, propose this as a practical construction of a microscope. It is intended simply and solely to show the fallacy of the supposed limit of 2 y assigned to the aperture, within a medium, of a pencil which can be brought without sensible aberration to a focus in air. As the sphericity rather than spheroidicity of the surfaces employed does not enter in any way into the arguments by which the limit in question is attempted to be established, the spheroidal or cartesian surfaces are quite admissible in argument.

Nevertheless, as an example of what can be done without going beyond spherical surfaces, and as bearing in a very direct way on actual practice, I will take another instance.

Let it be required to make a pencil diverging from a radiant point Q in glass diverge from a virtual focus q after a single refraction into air.

If P be a point in the required surface, μ QP - 9 q P must be constant, which gives, according to the value we arbitrarily assign to the constant, an infinity of cartesian ovals, any one of which, by its revolution round Qq, would generate a surface which may be taken for the bounding surface of the glass. In one particular case the oval becomes a circle, namely, when the constant = 0, in which case we have a circle cutting the line Qq internally and externally in the ratio of 1 to μ.

This case is represented in Fig. 1, in which O is the centre of the circle HAL, which by revolution round the line qQA generates the sphere. Rays diverging from Q within the glass proceed after refraction at the surface of the sphere as if they came from q. To find the limit of the pencil, we have only to draw the tangents q H K, q L M, and H K, L M will be the extreme rays after refraction. The incident rays Q H, QL corresponding to these are inclined to the normals O H, O L at the critical angle. It is easy to prove (as will appear from the postscript) that the lines QH, QL are prolongations of each other, so that the aperture in glass of the pencil which, after refraction into air, diverges without aberration from q is 180°. The aperture Hq L of this pencil, after refraction into air, is 2 y, which with the above value of y, for which the figure is drawn, comes to 81° 58'. Setting aside chromatic variations, the refracted rays proceed, of course, as if they came from q, forming

a pencil of aperture 81° 58'. A pencil of aperture in air no greater than 81° 58' is one which all parties allow can practically be brought to a focus; it could be brought exactly to a focus by the use of surfaces other than spherical.

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The spherical surface of no aberration accords with the form of the first lens to which the makers of immersion objectives have been led. By reducing somewhat the excessively large segment of a sphere represented in the figure, say reducing it to a hemisphere, the space gained in front (of thickness QO if the reduction be to a hemisphere) is available for the cover or interposed balsam, which have both nearly the same index as the crown glass of the first lens and the aperture in glass, though reduced from the extreme of 180°, still remains very large.

P.S.-The property of a circle employed in Fig. 2 admits of being proved in a few lines, and it might be convenient to the reader to have the demonstration.

Let O (Fig. 2) be the centre of a circle of which A B is a diameter. In OB take a point Q, and in O B produced take a point q so that Oq is a third proportional to OQ and the radius. Let R be any point in the circumference, and join QR, q R, OR.

Since the radius is a mean proportional between OQ and O q, we have in the two triangles QOR, qOR, which have a common angle at O, QO:OR::OR: 0q. Therefore the triangles are similar, and QR:qR::OQ:OR; and also the angles OQ R, Oq R are equal to qRO, QRO, respectively. Hence

Sin. q RO: sin. QRO:: sin. RQO: sin. Rq0qR: QR: OR: OQ. If then Q had been taken so that OQ: OB:: 1:μ, where μ is

the index of refraction of a sphere of which O is the centre and OB the radius, a ray QR proceeding from a point Q within the medium would after refraction proceed along q R produced. The limiting position of R is when qRO is a right angle, or q R a tangent to the circle, which is when RQO is a right angle, since then the sine in the angle of incidence QR: RO=1:μ, so that QRO is the critical angle.

FIG. 2.

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V.-On the Results of a Computation relating to Tolles' Objective. By Professor R. KEITH.

(Read before the ROYAL MICROSCOPICAL SOCIETY, June 5, 1878.)

HAVING received from Mr. Tolles, at the request of Mr. Mayall, jun., the elements * of the -inch immersion objective made by him for Mr. Frank Crisp, I have made a computation of its angular aperture (Plate VII.), and present a figure accurately representing its different lenses and their distances apart, and also the path of a ray of light, emanating from a focal point 10 inches behind the objective, and coming to a conjugate focus, free from aberration, 0·01620 of an inch before the front lens.

As a result of the computation, I find that when the objective is used with the thickest cover possible making balsam contact with the front lens, its aperture is 110° 11' 40". Under the same conditions the focal distance of the outside rays is 0.01620 of an inch, and of those near the axis 0.01618 of an inch, showing practically no aberration. The computation is made with more precision than is warranted by the nature of the elements, which are necessarily given to only two or three places of decimals, accurate enough, however, for the main purpose.

As the objective is thus shown practically free from aberration at the same time that its balsam aperture is far beyond that which corresponds to 180° of airangle, the only impropriety in calling its air-angle 180°, for the purpose of comparison, is in the fact that such a statement does not do justice to the objective.

With the computation Professor Keith wrote:-"Mr. Tolles has been liberal of his time in making the elements sure in every point; going so far as to make, from his memorandum in the case of Mr. Crisp's lens, a new objective in order to be more sure of the distances of the lenses apart and the final focal distance. These are not necessary as elements of the computation, but afford a very decisive confirmation of the correctness of the figuring."

The use of water instead of balsam, and perhaps the "setting" of the lens next behind the front, will slightly reduce the angle above given, but not by any material amount.

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The objective, as will be seen in the figure, is composed of seven lenses, Lettering them in order, from the back towards the front, the elements, as given by Mr. Tolles, are as follows:

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The distances between c and d and between ƒ and g are given by Mr. Tolles as very small, but not recorded in his memorandum; I have taken for the first 0.01 and for the second 0.006 of an inch. The adjustable distance between e and f Mr. Tolles gives as 0.07 when adjusted for a cover 0 014 of an inch thick. I have used 0.065 of an inch, which slightly increases the aperture, and is theoretically correct for balsam contact, or a cover 0.016 of an inch thick.

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