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Since this expression for Z is the sum of two squares, Z can never = 0, and therefore the six rays cannot be interrupted by a perfectly black band or ring. Perhaps, however, there may be a ring of light of such feeble intensity, interrupting the rays, as to appear like a black band cutting off the rays from the central part in the manner described by Sir J. Herschel. To ascertain if this be the case, let us find the value of m which gives Z a minimum. By differentiation we obtain

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The only factor in this expression which can be equated to zero, for the purpose of finding the maximum and minimum values of Z, being an exact square, Z admits neither of a maximum nor minimum, but decreases perpetually from the centre of the screen, as the following Table will shew.

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3. Let us now examine the part of the screen which is intermediate to two rays. The brightness of this part will be obtained by writing 30 for 9 in equation (B); which by that means becomes

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When m = 0, Z = 1; and as m increases Z diminishes ; at first rather slowly, but afterwards rapidly, so that when

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a = 1, or = r v3 = Hä. nearly, there is perfect blackness; some time before this, however, the light will be too feeble for vision, and there the light on the screen will appear to terminate unless the star or original luminous point of light be very bright, for as m goes on increasing, Z never attains a value

so great as 400 From this it follows that if there be a black ring surrounding

the central disc of light as described by Sir J. Herschel, its radius

will be such that “ yo - *: nearly. By reference to the above

Table, we perceive that the six rays have, at that distance from the

centre, a brightness of about # , the central brightness being represented by unity. If this be considered sufficiently feeble to constitute a black ring, we

are at a loss to aecount for the prolongation of the six bright rays mentioned by Sir J. Herschel, since their intensity has been shewn to decrease from the centre; and therefore when at any point in them the light is too feeble for vision, at every more distant point the light is still more feeble. Hence it would appear that, according to theory, the six rays are not interrupted by a dark ring, or band, in any part. In this particular, therefore, there is a decided disagreement between theory and the experiment recorded by Sir J. Herschel.

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4. Let us now examine the intensity in that part of the screen which is situated between any two of the six rays. As we have already seen the results when 0 = 0, and when 6 = 30, we shall now suppose the values of 6 to lie between 0° and 30".

is a factor of equation (B), it follows that the general

- I Since m'sin 36

brightness of the spectra will decrease very rapidly from the centre; and at a given distance from the centre the brightness is less the more 6 differs from 0, and is least when 6 = 30'. The places and extent of the spectra are pointed out by the other factor of equation (B), viz. sin' (m sin 0) sin’ {m sin (60' + 6); + sin' 3 m . (60 9); sin 6 sin (60' + 6) sin (60 – 9) “”

This expression vanishes entirely whenever m and 0 are such that (m sin 6) and (m V3 cos 6) are simultaneously both odd or both even multiples of t. If M be such a point, and MG, MH be drawn parallel to OA, OB, two of the six rays, then the distance of HM from OA, and the distance of MG from

/ON || OB are both multiples of (*) . Hence 0. \ there will be an infinite number of perfectly dark spots situated in the farther

corner of parallelograms, such as HMGO, whose sides are parallel to OA, OB.

If the line HM be such that its distance from OA is an even multiple

bX - - - - - - of a ’ then for every point in that line, the principal factor in the expression (C) is sin" s /// yo COS 0). which denotes spectra of the same charac

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ter as are exhibited in Fraunhofer's gratings.

If the distance of HM from OA be an odd multiple of o, then

the principal factor in the expression (C) is cos" (oys COS 9), which

represents spectra of the same kind as before, but intermediate to them

in position,

For a given value of m, r is greater for red than for violet coloured light, and consequently the spectra will have their red ends outwards, that is, farthest from the centre of the screen.

What is here said of MH referred to OA, is equally true of GM referred to OB: and what is said of the portion of the image within

AOR, is true of the portion within BOR; the line OR bisecting the angle AOB.

In Sir J. Herschel's experiment no spectra of this nature were seen, but with strong sun light they are very distinctly visible, and form to the six bright silvery rays a very beautiful appendage. In fact, on account of the remarkable symmetry of its parts, and of the great extent and extreme narrowness and whiteness of its six principal rays, which stretch completely across the field of view ; and on account of the number and geometrical arrangement of the coloured spectra, this experiment is inferior in beauty and splendour to very few of all those that have been exhibited in illustration of the science of Physical Optics.

S. EARNSHAW,

XXII. On the Decrement of Atmospheric Temperature depending on the Height above the Earth's Surface. By the Rev. J. CHALLIS, Plumian Professor of Astronomy and Eaperimental Philosophy in the University of Cambridge.

[Read February 13, 1837.]

THE temperature at any height above a given place on the Earth's surface is here considered to be the mean which would be found by a great number of thermometrical observations, made at that elevation, for a time sufficiently long to eliminate the diurnal and annual variations and the more irregular changes from winds. This mean temperature, it is known, varies with the height, and the object of this Paper is to enquire respecting the law of the variation.

The causes which determine the temperature of the atmosphere at a given elevation, are probably of a very complicated nature, but among the principal may be reckoned the diminution of density in the higher regions. In the following reasoning it is assumed, that the temperature and density are functions of the height, and the effect of decrease of density will be considered apart from every other circumstance. If then 6 be the temperature and p the density at the height x, we shall have

d6 - d6 d6 dp (#)-####, (1). Vol. VI. PART III. 3 L

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