Then, the rays which diverge from E and fall upon AB, will, after reflection, diverge from G, that is, G will be an image of E. Also, these rays, after reflection at AB, will fall upon CD as if they proceeded from a real object at G, and after reflection at CD they will diverge from H; that is, H will be an image of G, or a second image of E, &c. In the same manner, the rays which diverge from E, and fall upon CD, will form the images L, M, N, &c.

It is found, by experiment, that all the light incident upon any surface, however well polished, is not regularly reflected from it. A part is dispersed in all directions, and a very considerable portion enters the surface, and seems to be absorbed by the body. In the passage also of light through any uniform medium, some rays are continually dispersed, or absorbed; and thus, as it is thrown backward and forward through the plate of air contained between the two reflectors, AB, CD, its quantity is diminished. On all these accounts, therefore, the succeeding images become gradually fainter, and, at length, wholly invisible.

Cor. If E move towards F, the images G, H, I, &c. move towards the reflectors, and L, M, N, &c. from them; thus the images L and H, M and I, respectively approach each other, and when E coincides with F, these pairs respectively coincide.

PROP. 14.-If an object be placed between two plane reflectors inclined to each other, the images formed will lie in the circumference of a circle, whose centre is the intersection of the two planes, and radius the distance of the object from that intersection.

=

Let AB, AC, fig. 22, be two plane reflectors, at the angle BAC, E an object placed between them. Draw EF perpendicular to AB, and preduce it to G, making FG EF; then the rays which diverge from E and fall upon AB, will, after reflection, diverge from G, or G will be an image of E. From G, draw GH perpendicular to AC, and produce it to I, making HIGH, and I will be second image of E, &c. Again, draw ELM perpendicular to AC, and make LM EL; also, draw MNO perpendicular to AB, and make NO MN, &c. and M, O, &c. will be images of E, formed on the supposition that it is placed before AC. Let K, V; P, Q, be the other images, determined in the sam

Then, since EF is equal to FG, and AF common to the triangles AFG, AFE, and the angles at F are right angles, AG is equal to AE (Enc. 4. 1). In the same manner it appears, that AI, AK, &c. AM, AO, AP, &c. are equal to each other, and to AE, that is, all the images lie in the circumference of the circle EMIK whose centre is A and radius AE.

Cor. If the angle BAC be finite, the number of images is limited. For, BA and CA being produced to S and R, the rays which are reflec

ted by either surface, diverging from any point Q between S and R, will not meet the other reflector; that is, no image of Q will be formed.

PROP. 15.-Haring given the inclination of two plane reflectors, and the situation of an object between them, to find the num ber of images.

It appears from the construction in the last proposition, that the lines EG, MO, IK, PQ, &c. are parallel, as also EM, GI, OP, KV, &c. Hence it follows, that the arcs EG, MI. OK, PV, &c. are equal; as also, the arcs EM, GO, IP, KQ, &c. Let BC=«, EB = b, EC=e; then the arc EG 26, EM = 2c, EO= EG + GO≈ EG + EM=2b+2c =2a, EK=EÓ + OK = EO + EG = 2a +2b, EOQ= EK + KQ =EK+EM=2a+2b+2c4e, &c. Thus, there is one series of images, formed by the reflections at AB, whose distances from E, measured along the circular arc EOR, are 2b, 2a + 2b, 4a+2b, ..... Qua −2a+2b (2na — 2c), where n is the number of images, this series will be continued as long as 2na - 2a+2b, or 2na-2e is less than the are EOR, or 180°+b, and consequently n, the number of images in this 180+b+2c series, is that whole number which is next inferior to 2a 180+a+c. There is also a second series of images, formed by reflec tions at the same surface, whose distances from E are 2a, 4, 64, 2ma, continued as long as 2ma is less than 180 + b, and there forem, the number of these images, is that whole number which is next

2a

......

inferior to

180+b

2a

or to

In the same manner, the number of images formed by reflections at the surface AC, is found by taking the whole numbers next inferine to 180+a+b 180+c

be

2a

and

2а

Cor. 1. If a be a measure of 180, the number of images tormed will

360

a

For, if a be contained an even number of times in 180, or 2a be a measure of 180, the number of images in each series is ; and the

180 $60
2a

180
2a

number upon the whole is 4 X =--- If a be contained an odd

a

number of times in 180, 2a is a measure of 180+a, or 180

180+ a 180 at
+

the number of images is

2a

60

a

2a

Cor. 2. When a is a measure of 180, two images coincide.

SUP

For, if a be contained an even number of times in 180, then the number of images in the second series, formed by reflections at the surface 180 AB, is ; and the distance EOQ (2ma), of the last image from E, is 2a 180°. In the same manner, the distance EIV, of the last image, in the second series, formed by reflections at AC is 180°, therefore the two images, Q and V, coincide in EA produced. If a be contained an odd number of times in 180, then the number of images in the first series, 180+a formed by reflections at AB, is and the distance EOK of the " Qu

last of these images from E, is

180+a
2a

X 2a-2c; or 180° + a2c.

Also the distance EMP, of the last image in the first series formed by reflections at AC, is 180° +a—26; therefore EOK+EMP=360° +2a-2c2b=360°, that is, K and P coincide.

ASTRONOMY.

THE

LAWS OF CENTRIPETAL AND CENTRIFUGAL FORCE: Showing the Motions of Bodies in Circular Orbits, and

in the Conic Sections, and other Curves.

IN the following chapter is explained and demonstrated the laws of Centripetal Forces; a doctrine upon which all Astronomy is grounded, and without the knowledge of which no rational account can be given of the motions of any of the celestial bodies, as the Comets, the Planets, and their Satellites.

In the first section are given the centripetal forces of bodies revolving in circles; their velocities, periodic times, and distances, compared together; their relations and proportions to each other, and that when they either revolve about the same centre, or about different ones. The different motions caused by different forces, or by different central acting bodies, are here shown.

In the second section are shown the motion of bodies in the ellipsis, byperbola, and parabola, and in other curves; the proportion of the centripetal forces, and velocities in different parts of the same curve; the law of centripetal force to describe a given curve, and the velocity in any point of it; and more particularly with respect to that law of centripetal force that is reciprocally as the square of the distance, which is the grand law of nature in regard to the action of bodies upon one another at a distance; and accordiing to this law is shown the motion of bodies round one another, and round their common centre of gravity, and the orbits they will describe.

Definitions.-1. The centre of attraction, is the point towards which any body is attracted or impelled.

2. Centripetal force, is that force by which a body is impelled to a certain point, as a centre. Here all the particles of the body are equally acted on by the force.

3. Centrifugal force, is the resistance a moving body makes to pre. vent its being turned out of its direct course. This is opposite and equal to the centripetal force; for action and re-action are equal and contrary.

4. Angular velocity, is the quantity of the angle a body describes in a given time about a certain point, as a centre. Apparent velocity is the same thing.

5. Periodical time, is the time of revolution of a body round a

centre.

J. On the Motion of Bodies in Circular Orbits.

1. The centripetal forces, whereby equal bodies, at equal distances from the centres of force, are drawn towards these centres, are as the quantities of matter in the central bodies.

For, since all attraction is made towards bodies, every part of the attracting body must contribute its share in that effect. Therefore, a body twice as great will attract the same body twice as much; and one thrice as great, thrice as much, and so on. There fore, the attraction of the central body, that is, the centripetal force, is as the quantity of matter in the attracting or central body. 2.-Cor. 1. Any body, whether great or little, placed at the same distance, is attracted through equal spaces in the same time, by the central body.

For, though a body, twice or thrice as great as another, is drawn with twice or thrice the force, yet it will acquire no greater velocity, nor pass through a greater space. For (by Mechanics,) the velocity generated in a given time is as the force directly, and quantity of matter reciprocally; and the force, which is the weight of the body, being as the quantity of matter, therefore the velocity generated is as the quantity of matter directly, and quantity of matter reciprocally, and therefore is a given quantity.

3. Cor. 2. Therefore, the centripetal force, or force towards the centre is not to be measured by the quantity of the falling body, but by the space falls through in a given time. And, therefore, it is sometimes called an a celerative force.

4.-If a body revolves in a circle, (fig. 1.) and is retained in it, by a centripetal force tending to the centre of it; put R = radius of the circle or orbit described, AC.

Fabsolute force, at the distance R.

the space a falling body could descend through, by the force at A; and

t = time of the descent.

3.1416.

Then its periodic time, or the time of one revolution will be
And the velocity, or s ace it describes in the time t,

2R

will be Rs.

For, let AB be a tangent to the circle at A; take AF an infinitely small arch, and draw FB perp. to AB, and FD perp. to the radius AC. Let the body descend through the infinitely small height AD or BF, by the centripetal force in the time 1. Now that the body may be kept in the circular orbit AFE, it ought to describe the arch AF in the same time 1. The circumference of the circle AE is 2R, and the arch AF = √2R × AD.

By the laws of falling bodies

AD st: √AD: t√ = time of moving through AD or AF. And, by uniform motion, as AF, to the time of its description :: circumference AFEA, to the time of

one revolution; that is, √2R × AD :t

AD

:: 2R: periodic

Also, by the laws of uniform motion, & AD,

or time of describ

ig AF: AF or ✓/2R × AD ::t: √2Rs = the velocity of the body, or space described in time t.

5.-Cor. 1. The velocity of the revolving body is equal to that which a falling body acquires in descending through half the radius AC, by the force at A uniformly continued.

Fors (height): 28 (the velocity) :: ✔R (the height): √2Rs, the velocity acquired by falling through R.

6.-Cor. 2. Hence, if a body revolves uniformly in a circle, by means of a given centripetal force, the arch, which it describes in any time, is a mean proportiona! between the diameter of the circle and the space which the body would descend through in the same time, and with the same given force.

For 2R (diameter): √2Rs :: ✔Rs; s; where 2K is the arch described, and s the space descended through, in the time t.

7.-Cor 3. If a body revolves in any curve AFQ, (fig. 2,) about the centre of force s; and if AC or R be the radius of curvature in any point A ; &= space descended by the force directed to C; then the velocity in A will be V2Rs.

For this is the velocity in the circle; and therefore in the curve, which coincides with it.

8.-If several bodies revolve in circles round the same or different centres, the periodic times will be as the square roots of the radii directly, and the square roots of the centripetal forces reciprocally.