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if n, ng ng are all functions of N, making the latter the independent variable and dividing by dN, we have
also the dispersion of a ray after passing m surfaces, or
This formula by successive substitutions may be applied to any case. For a single surface
m=1,1, = 0, and hence
which equals unity when tang r=n, or at the angle of total polarization. That is, the unit of dispersion is that produced by a single surface when the ray is in the position of total polarization. For two surfaces (4) becomes
in a prism ng = - making suitable substitutions, and reducing we obtain
a being the angle of the prism. For minimum deriation
this does not, however, give the minimum of dispersion for which the condition (found by differentiating) is
no sin (a + r) cos (a + 2 r,) + sin r1=0 (5)
an equation which has, I believe, never been solved. When a is very small
sin (a + r) =a + ru, cos (a + 2 r) = 1 sin r =r
I find that this equation gives very nearly the true value of rı, even when a is large, thus for n = 1.5, a = 15°, 30°, 45°, and 60° it gives r = 10° 23', 20° 46', 31° 9', and 41° 32', while the true values obtained by trial in (5) were 10° 23', 20° 45', 31° 6' and 41° 29'.
To show the comparative deviation and dispersion, while a prism is rotated, I have calculated the following table for prisms whose index of refraction is 1.5, and their angle 15°, 30°, 45°, and 60°; r, is the angle at which the light leaves the prism, and i, that at which it enters. The columns headed “ Reduced Deviation” are calculated by the condition that for small angles at which the dispersion and deviation are proportional their units shall be the same.
From this table it follows that if we turn the prism of a spectroscope 20° from minimum deviation, the dispersion will be nearly doubled, thus doubling the power of the instrument. In projecting the spectrum on the screen, this device is often useful; I have thus, by a single 60° prism, projected a spectrum of such a size that the two parts of the D line were about a millimetre and a half apart, each having a thickness of only about half a millimetre.
Since the dispersion does not depend strictly upon the deviation, and we may have two prisms producing the same dispersion but unequal deviations, we can evidently make one achromatize the other, even if they are made of the same kind of glass, and have the same angle. Thus if two 15° prisms be so placed that r, shall be 50° for one, and 37° 21' for the other, the dispersions of each will be .418, while the deviation will be 11° 7' for one, and 17° 53' for the other, as may be seen from the table. Again, if the ray of light passes through first one and
then the other, the second will neutralize the dispersion of the first, while there will remain 17° 53' — 11° 7' = 6° 46' deviation. It was by this arrangement of prisms that Brewster obtained achromatism with one kind of glass.
Perhaps the most important application of these principles is to photographing the spectrum. Photographs taken by the common methods are greatly distorted, particularly in the more refrangible end. They VOL. VII.
can only be used by identifying the more prominent lines, with those whose wave-length is known, and interpolating the remainder approximately. If, as is often the case, these standard lines cannot be recognized, the photograph becomes useless. To show the amount of distortion, suppose a spectrum to contain three similar double lines A, B, and C, whosé indices of refraction are 1.5, 1.6, and 1.7, and that we use a 60° prism, the line B being in the position of minimum deviation. The deviations of the three lines will then be 48° 34', 53° 8', and 58° 11', and their dispersion 1.528, 1.667, 1.878; that is, A and C will be at distances of 4° 34' and 5° 3' from B instead of equidistant, and the interval between the components of each line will be as 1.528: 1.667 : 1.878; the distortion in this case amounting to about 20 per cent.
If now the portion of the screen which receives the line C be brought nearer the prism, the parts of this line will approach one another, and since their distance apart is proportional to their distance from the prism, the three lines will appear alike, if the screen is so inclined that the points where they are projected are at distances,
A simple calculation shows that the screen must be slightly curved to fulfil this condition, but if plane, the distortion will be only about one and a half, instead of twenty per cent, the angle of inclination with the ray B being about 37°. If an achromatic lens was used for the projection, all parts of the spectrum would not be in focus, but with a single lens the focal distance of the violet is always less than that of the red rays. If then we use such a lens, inclining the screen at the same time corrects the distortion, and brings all parts into focus at once. By placing the prism at a suitable distance from the lens, both sources of error may be almost entirely eliminated. The oblique incidence of the light on the sensitive surface would be an objection to this method, but would be partly counterbalanced by the fact that the length of the spectrum would be thereby increased more than one half. Or, if preferred, the prism may be turned, so that, applying the correction for distortion as above, the screen shall be more nearly perpendicular to the light.
In conclusion, this spectrum would possess the following advantages over the distorted forms now in use. Horizontal distances being proportional to the change in the index of refraction, the latter could be determined at once for any line, by a scale of equal parts. Its extent would be much greater than that of the visible spectrum, and we could determine the index of refraction of rays of too short wave-length to be measured readily by the common methods. It would be a normal spectrum for any given material, being independent of the form and position of the prism. And (especially if the interference bands were produced in it) it would afford, from its extent, great advantages for the study of the laws of the dispersion of light by different substances.
Five hundred and ninety-fourth Meeting.
May 12, 1868. — MONTHLY MEETING. The PRESIDENT in the chair.
The Corresponding Secretary read letters relative to exchanges ; also a letter from Professor De La Rive in acknowledgment of his election into the Academy as a Foreign Honorary Member.
Mr. C. M. Warren presented by title a memoir on “ Volatile Hydrocarbons in Pennsylvania Petroleum."
DONATIONS TO THE LIBRARY,
FROM JUNE 2, 1865, TO JUNE 30, 1866. State of Massachusetts.
Report to His Excellency the Governor and the Honorable Council, of the Commissioners appointed under the Resolve of May 3, 1865, “concerning the Obstructions to the Passage of Fish in the
Connecticut and Merrimack Rivers.” 8vo pamph. Boston. 1866. Massachusetts Historical Society.
Proceedings. 1864 – 1865. 8vo. Boston. 1866. Massachusetts Institute of Technology.
First Annual Catalogue of the Officers and Students, and the Programme of the Course of Instruction, of the School of the Massachusetts Institute of Technology. 1865 – 66. 8vo pamph. Boston. 1865.