pasteboard when they illuminate their respective halves of the tissue paper equally. Let then the object-glass C be interposed between Q and fn; the eye will now see the illuminations on the tissue paper to be unequal, and the other flame will require to be moved to some new position as 92 to make them again equal. The position of 92 will depend on the condensing power of the lens, and on the portion of light lost in passing through it; the former effect being calculated from the focal length and the position of Q, the latter is determined from the results of the experiment.

To find the intensity which the transmitted light would have had if it had not been condensed into a smaller area by the effect of the lens, let Q in the annexed figure represent the position of

a

A

d

radius

the flame, C that of the lens, and AdB the place of the screen; let QC-u, Cd=d, f= the focal length of the lens, and a of its aperture. The light diverging from Q which falls on the lens, would have illuminated a circular area AB if the lens had not been interposed, but when emergent from the lens it will diverge from the focus q, which is conjugate to Q, and illuminate the circular area ab. If qC=v we have (PART I. Art. 65.)

and, referring to the first figure, we have from the experimental result

intensity of the light condensed on ab___ (q1m)2
intensity of the light direct from Q

Compounding with the previous result we have

intensity of the transmitted light if not condensed

intensity of the direct light

=

=

2

By this method a very fine double achromatic object-glass of four inches aperture and six feet focal length was found to transmit about sixty-six rays out of every one hundred incident. As the apertures of object-glasses of telescopes are increased, the thickness of the lenses must be increased also, and hence the loss of light will be greater than in smaller ones. If the crown or plate glass of which the convex lens is made has much colour, the loss of light will be much increased.

ART. 4. PROBLEM. To compare the intensity of the light of the heavenly bodies beyond the limits of the atmosphere, with that at the earth's surface.

To solve this problem rigorously we should have given the density of the atmosphere at all altitudes to which it extends, and the portion of light transmitted (differing with different seasons and

climates) through given spaces for any given density, but we may expect to obtain a good approximation by taking the height to which the atmosphere would extend, if its density, at all altitudes, were the same as at the earth's surface, at the mean; and supposing that light is transmitted through this homogeneous atmosphere as it is at the earth's surface.

Bouguer concluded from his experiments, that light was diminished in the ratio 2500 to 1681 in passing through 7469 toises (9.046 English miles) of dense air; which was nearly the ratio of the sun's light traversing the atmosphere at the summer and winter solstices at Croisic, his place of residence.

To apply this to the formula I=IM' of Art 2, with t expressed in miles, we have

Supposing that the earth's atmosphere, if homogeneous, would extend to the height of 4.9 miles, and I were the intensity of light from a heavenly body beyond that limit, we have for the intensity at the earth's surface, after traversing the atmosphere vertically,

14.9=IM4.9
=I(.80655)

or nearly one-fifth of the light traversing vertically the earth's atmosphere is absorbed, or dispersed.

To find the proportion absorbed when a heavenly body has any given zenith distance, we must find the distance t through which the light passes within the homogeneous atmosphere, and apply it in the preceding formula.

Let O be the center of the earth, AA' the surface, z the zenith of the place A, in the radius OA produced, Aa=the height of the

then AO the radius of the earth being known; OB=Oa, and the zenith distance z AB being

also known, we have AB=t by solving the triangle OAB.

I

The formula

=

I

Mt
M

=M(t−") gives the ratio of the intensities

for two different altitudes when t and t have been found.

CHAPTER II.

ON THE REFLEXION OF LIGHT BY PLANE AND CURVED MIRRORS.

In this chapter we have to discuss the higher propositions of Catoptrics which were omitted, or only mentioned, in PART I.

ART. 5. PROP. To find the direction of a ray of light after being reflected by two plane mirrors, the planes of reflexion being any whatever.

Let SA, AB, BT be the course of a s ray which is reflected at A by the mirror LM, and at B by the mirror KO; NA, nB being the normals to the mirrors respectively.

If SA were one of a pencil of parallel rays, the reflected rays would be all parallel to AB; and similarly after the second reflexion each ray would be parallel to BT; so that the deviation after the two reflexions would be the same for each ray, and independent of the points of incidence on the two mirrors. Let us suppose that A is indefinitely near to O, and that SO in the lower figure represents the incident ray, ON the normal, and ROr the direction of the reflected ray. Let On parallel to Bn represent the normal to the second mirror, and

N'

OT the direction of the ray after being twice reflected.

Let a sphere with radius unity be described round O, and let the lines above-named be radii, the plane of the first reflexion