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sines, and i for the common sine of incidence; for we shall have as before v➡ i to ri in a given ratio; which call that of 1 to q. And from these two, we
have = = = = = × 1. But p is always nearly equal to q; in the refractions
from glass, and from water into air, their difference is less than
part of either; we may therefore put the ratio- equal to which is the first theorem.
And thence, if one difference R-1 become equal to ri, the other differences v-1, &c. will be respectively equal to v―i, &c. and the same set of differences may be made to serve for several media, provided the sines of incidence are taken in their due proportion. Thus when red rays pass from glass into the air, we have I: R:: 50: 77 and R-1:1:: 27: 50, and when they pass from water into air i : r — i :: 3 : 1, and therefore, as we are to make R― I every where equal to r - i, we get, ex æquo, i: 1 :: 81: 50, as Sir Isaac Newton finds it.
Remark 4. But to explain this matter a little further, and obviate some difficulties concerning it, I shall add the following examples.
The refractive powers being marked as above, let red rays fall from glass into air at the angle of incidence 20°, the angle of refraction will be 31° 47'. Again, let them fall from water into air at an angle of 34° 1′, making their angle of refraction 48°5'. And the difference of the sines of 31° 47′ and 20° will be precisely equal to the difference of the sines of 48° 5' and 34° 1'. At the same angles of incidence 20° and 34° 1′, let the violet rays fall from glass and water into air, and the angle of refraction from the glass will be 32° 14' nearly, and that from the water will be 48° 38′ nearly. And the difference of the sines of 32° 14'4 and 20° will be equal to the difference of the sines of 48° 38′ and 34° 1′, within .000488, or less than part. We see likewise that the red and violet rays diverged from the glass medium at an angle of 27'; but from the water at an angle of 33′; making the difference of divergence in this example 5', that is of the whole divergence of the red and violet rays when refracted from glass into air, at incid. 20°. Whence it appears, that though the differences of the sines above specified, or the excesses in Sir Isaac's theorem, may in refractions from different media into the same rarer medium, be made equal, it does by no means follow, that the divergences of the several sorts of rays (or if you chuse to call it their dispersion) will be the same in the 2 refractions; for Sir Isaac's excesses 27, 27, &c. are the excesses of sines; not of angles, as some opticians seem to have misapprehended.
Again, let an unrefracted pencil of light fall from common glass into the air, fig. 7, at the incidence 39°, and the angles of refraction will be,
And their difference.... 3 18 7 is the divergence of
the extreme rays. And the angle of refraction of the mean ray is 77° 16′ 19′′. By mean ray is understood the ray whose sine of refraction is a geometrical mean between the sines of refraction of the extreme rays, the common radius being unity.
Let now the same rays be refracted the contrary way by a surface of water WT; then, to make the mean emergent ray parallel to the incident pencil, its angle of incidence must be 86° 37' and the extreme rays will now converge at an angle of 20 minutes, nearly. Through the point of convergence o, draw (by the Lemma) a plane wt, to terminate the water, and unite all the rays into a colourless pencil os: and this emergent pencil will be found to make, with a perpendicular to the terminating surface, an angle of 49° 6+, and will be inclined to the first incident pencil in an angle of 14° 28′ 20′′. Nor is there any other plane besides this which will thus unite the rays. If planes parallel to it cut the rays any where but in their point of convergence, they will be parallel to each other, but exhibiting their several colours. And planes not parallel to it, will every where give a coloured image, excepting only when they pass through the point of convergence; but then the rays having crossed at that point, will.. thenceforth diverge from one another, and give a coloured spectrum..
From all which it appears, that light refracted through different media may emerge colourless, though its first direction be considerably altered. And that its mean direction may remain the same, though its extremities be sensibly. tinged with colours. Positions which, I know not by what mishap, have been deemed paradoxes in Sir Isaac Newton's theory of light, but which are really the necessary consequences of it.
Of Telescopical Object-Glasses giving an Image free from Colours. Fig. 8 and 9. If the extreme rays, the red and violet, after one or more refractions, diverge from points D and d, the distance of the point of divergence of the least refrangible from the lens, being greater than that of the most refrangible, such a semidiameter of the last spherical surface, from which they are to pass into the air, may be assigned, as shall unite the extreme, and all the intermediate rays, in the same focus F; neglecting the aberration from the figure. The Rule is this: For the distances of the points of divergence from the lens, write D the greater, and d the least; the semidiameter of any of the given surfaces being assumed for unity: and, expressing the ratios of the sines of incidence and refraction, of the violet and red rays, out of air into the last medium whose surface is required; the semidiameter of that surface will be (Mm) x Dd ; as may be easily demonstrated from a theorem of Dr. Smith, in the remarks subjoined to his Optics. Thus, if the last medium be glass, the semidiameter of the sun
face from which the rays pass into the air, must be
it being, in this case, 78D 77d' EXAMPLE 1.-Let мpсNCм, fig. 8, be a double convex lens of water, confined between the plano-concave MTLN, and the meniscus MKNCM, both of glass, and having the radii of their surfaces contiguous to the water, equal to each other, or to unity; and if a ray sp, parallel to the common axis of the lenses, after being refracted by the aqueous lens, have its extreme rays, the red and violet, divergent from the points D and d; the distance of r, the focus where all the rays can meet, will be 8.898: and when this happens, the exterior surface of the meniscus, that is, the surface represented by MPKN, will have its radius to that of the inner surface MCN, as 139 to 154.
EXAMPLE 2.—When a double concave of glass, the radii of whose surfaces are unity, is inclosed in water, as in fig. 9, the water being confined on one side by a thin glass plate TL, and on the other by a concentric spherical shell MPKN; the semidiameter of this shell must be to unity as 471 to 547; and the focal distance cr, at which the colourless image is formed, will be 4.77. In these examples, the thickness of the lenses is neglected; but it may easily be taken into the account, if it be thought necessary.
The same thing may be effected by means of any media of different refractive powers; for the semidiameter of the last refracting surface being determined according to the foregoing rule, the nearer distance of the points of divergence (d), of the more refrangible rays, will be so compensated by their greater refrangibility, that all the rays will converge to the same focus F. And this without introducing any new principle into the science of optics, or any dispersion of light different from the refractions discovered by Sir Isaac Newton near a hundred years ago,
C. and R. Baldwin, Printers,
END OF VOLUME ELEVENTH.