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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 0 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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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

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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 m√3 18π will be such that nearly. By reference to the above 12

=

2

Table, we perceive that the six rays have, at that distance from the centre, a brightness of about 1, the central brightness being represented by unity.

If this be considered sufficiently feeble to constitute a black ring, we are at a loss to account 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.

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, and when 0 0 0, and when 0 = 30°, we shall now suppose the values of to lie between 0° and 30o.

1

Since m1 sin 30 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 differs from 0', and is least when = 30°. The places and extent of the spectra are pointed out by the other factor of equation (B), viz. sin2 {m sin (60° + 0)} sin2 {m. (60° – 0)} sin (60° - 0)

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

sin2 (m sin 0)
sin 0

sin (60+)

+

.....

.(C).

This expression vanishes entirely whenever m and are such that

(m sin e) and (m3 cos 0) are simulta-
neously both odd or both even multiples

of π.
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

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of

M

B

H

A

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

a

then for every point in that line, the principal factor in the ex

pression (C) is sin2 (√3 cose), which denotes spectra of the same charac

3 2

ter as are exhibited in Fraunhofer's gratings.

bx

If the distance of HM from ОA be an odd multiple of then

a

m√3
2

cos e),

the principal factor in the expression (C) is cos2 e), 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 40B.

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 Experimental 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 be the temperature and p the density at the height x, we shall

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de dz

in which equation expresses the variation of temperature corresponding to a change of height, so far as it varies independently of change of density. If also p be the pressure where the density is P, and g be the force of gravity, we have

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Lastly, we have the known relation between the pressure, density, and temperature, given by the equation

p = a2p (1 + a0), (3).

2

in which the temperature is supposed to be reckoned in degrees of the centigrade thermometer, a is the pressure where p=1 and 0=0, and a is the numerical coefficient 0,00375. With respect to the equation (2) we may remark that though it is in strictness applicable only to the air at rest, it is very nearly true when the atmosphere is in motion; for the direction of winds is necessarily nearly parallel to the Earth's surface, and consequently the effective accelerative force in the vertical direction is very small. Hence is nearly equal to the impressed accelerative force, that is, to the force of gravity.

The equation (3) differentiated gives,

dp pdz

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and by substituting this value of dp in (1), it will be found that

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dz

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a2 (1 + a0) dp

αρ de 1 + a0 dp

(4).

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