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REGISTER OF BAROMETER AND THERMOMETER, 1836.

(Computed from the REGISTER at the PHILOSOPHICAL SOCIETY.)

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October..... 29.79 30.44 28.88 1.56 .92 .00 .25
November... 29.60 30.12 29.05 1.07 .79 .01 .26 42.00°
December... 29.78 30.35 29.09 1.26 .67 .01 .21
For the year
29.83 30.65 28.85

June

July..

.....

August...... 30.03 30.35 29.58

September... 29.88 30.35 29.35 1.00 .52 .00 .14 53.90° 71° 36° 35° 26° 48.19°

6o 12.22° .49 .00 .18 44.95° 61° 32° 29° 23° 5o 13.98° .36 .00 .11 54.29° 79° 36° 43° 34° 12° 19.30° .34 .00 .14 62.43° 86° 46° 40° 34° 5° 18.38° .76 .49 .01 .15 63.77° 93° 43° 50° 36°

2

4

0

0

5o 19.00°

0

.77 .36 .00 .11|| 60.95° 80° | 45° | 40° 31° 10° 18.74°

0

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65° 28° 37° 22°
56° 29° 27° 23°

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39.38° 55° 24° 31° 20° 1.80 .92 .00 .18 49.16° 93° 19° 74° 36° 2° 14.12° 58

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Mean

Diurnal Range.

Number of Days

Minimum

at 32o or under.

XVII.

On the Intensity of Light in the neighbourhood of a Caustic.
By GEORGE BIDDELL AIRY, Esq. A.M., Astronomer Royal:
Late Fellow of Trinity College, and Plumian Professor of As-
tronomy and Experimental Philosophy in the University of
Cambridge.

[Read May 2, 1836, and March 26, 1838.]

WHEN a great physical theory has been established originally on considerations and experiments of a simple kind, which by degrees have been exchanged for comparisons of more distant results of the theory with more complicated cases of experiment, it has always been considered a matter of great interest, to trace out accurately by mathematical process the consequences, according to that theory, of different modifications of circumstances: which can which can then be compared with measures that have been made, or that may easily be made in future. It is with this view that I solicit the indulgence of the Society, for the following investigation of the Intensity of Light in the neighbourhood of a Caustic, as mathematically estimated from the Undulatory Theory.

The investigation which I present here belongs, ostensibly, only to the case of reflection. The introductory part of it will, however, (with the proper modifications) apply equally well to all cases of refraction and all combinations of reflection and refraction. There seems also to be no reason why the latter part (the estimation of the intensity of light, by considering the wave of light when it leaves the last surface to be divided into a great number of small parts, whose separate effects

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For

are then to be compounded,) should not apply to those cases. though, strictly speaking, we ought to consider the wave to be thus broken up where it leaves the first surface, in order to find the intensity of vibration at every point of the second; yet it seems clear, that those reasonings which establish the definite reflection or refraction of a wave, (and which are founded upon the consideration above alluded to,) point out that there will be as to sense a mutual destruction of all vibrations at the second surface, (supposed to be not distant from the first,) excepting those which would be fully taken into account on the ordinary laws of Geometrical Optics. Where the light meets the second surface in the state of convergence, this conclusion perhaps is not so clear: but even there I believe that it may easily be shewn to be correct. I have mentioned these points because one of the most interesting cases of natural caustics (the rainbow) is affected by them; the exterior bow involving the first-mentioned condition, and the interior bow involving both the first and the second.

1. The notion of a caustic, and its mathematical definition, are essentially founded upon the laws of Geometrical Optics; and to these, therefore, we must refer in order to discover a representation of the conditions adapted to the investigations of Physical Optics. For simplicity we shall confine our diagrams to the plane of reflection, and shall consider the reflecting surface as symmetrical (to a sensible extent) with respect to that plane, so that the portion of the caustic formed by that part of the surface will be in the same plane.

S be the origin of co-ordi

of the reflecting surface; reflected ray; the length

2. In fig. 1., let the origin of light nates; x, y, the co-ordinates of a point p, q the co-ordinates of a point P in the of the path of light from S to any point of the reflecting surface and thence to the point P. The ordinary law of reflection informs us that the angles of incidence and reflection are equal; and therefore, that, if we take a point X' on the reflecting surface very near to X, and join it with the origin and the point P, the lengthening ZX' of one

of these lines will be equal (ultimately) to the shortening XZ' of the other, and their sum (ultimately) will not be altered; or that, putting d(V) dx

for the differential coefficient of V with regard to a, considering dV dv dy y also as a function of x, (which is otherwise written +

d(V) dx

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dx dy dx' = 0. This is the condition which holds at the point of reflection.

3. Now if p, q, be the co-ordinates of a focus, since in that case it is a point in the paths of rays reflected from every point of the surd2 (V) = 0 at every point, and therefore V = = constant, and dx

face, d3 (V) dx3

d(V)
dx

&c. are = 0 at every point. This is the condition for the reflection of rays to a focus.

4. But though the condition V C and all its consequences are necessary for the convergence of reflected rays to a focus, yet this condition is not necessary for the convergence of a very small pencil of rays incident on the reflecting surface. It is only necessary for

this, that the equations

d(V)
dx

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same time, when x = x + dx and has the corresponding value; that is, that the following equations should be true at the same time,

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expressing that the rays incident at the points x and x + dx intersect: and making a indefinitely small, this reduces itself as nearly as we

please to the equation

d2 (V)
dx2

= 0.

This then is the equation which must hold for the ultimate convergence of rays.

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