of its two surfaces are equal, it is said to be equally convex; and when the radii are unequal, it is said to be an unequally convex lens.

5. A plano-convex lens, shown at E, is bounded by a plane surface on one side, and by a convex surface on the other.

6. A double-concave lens, shown at F, is bounded by two concave surfaces, whose centres are on opposite sides of the lens.

7. A plano-concave lens, shown at G, is bounded by a plane surface on one side, and a concave surface on the other.

8. A meniscus, or new-moon-shaped lens, shown at H, has one side concave, and the other convex.

9. A concavo-convex lens, shown at 1, is bounded by a concave surface on one side, and a convex surface on the other.

The axis of these lenses is a straight line, MN, in which the centres of the spherical surfaces are situated, and to which their plane surfaces are perpendicular. If the sections from в to I were to revolve round the line M N, they would generate the different solid lenses which they represent; but, in treating of the refraction of the lenses, we shall use the sections, because every section of the same lens passing through the axis M N has the same form, and hence what is true of one section must be true of the whole lens. The reader will bear in mind that the convex surface of a lens is like the outside of a watch-glass, and the concave surface like the hollow or inside of a watch-glass.

Refraction through Prisms.-As prisms are used in the construction of several optical instruments, and are essential parts of the apparatus employed for decomposing light and examining the properties of the com

ponent parts of the solar beam, it is desirable that the reader should be able to trace the progress of light through their two refracting surfaces. Let ABC be a prism of plate glass, having its refracting power 1.525, and let H R be a ray of light falling obliquely upon the

face A B at R. Through R draw M R N perpendicular to A B, and from any scale of equal parts take in a pair of compasses 1.525, or 15.25 parts, and, setting one foot of the compasses on H R, move it along to some point н till the other foot falls only on one point m of MR; then, with R as centre, and H R as radius, describe the circle н м b. From the same scale take in the compasses 1.000 or 10.00, and, setting one foot on the line R N, move it along to n till the other falls upon b in the circle Hмb, taking care that when one foot is placed at b, the other foot can touch RN in no other point but n. But Hm is the sine of the angle of incidence, and bn the sine of the angle of refraction; therefore the line Rbs, drawn through b, will be the refracted ray.

Again, as the ray Rbs meets the second refracting surface, AC at s, through s, draw NST perpendicular to A C, and from any scale of equal parts take in the compasses 1.000 or 1000, and, setting one foot on the line s b, move it along to some point o, till the other

falls only on one point of s N, as at P. In like manner take from the same scale 1:525 or 15.25, and, setting one foot of the compasses in s T, move it towards some point q, till the other foot falls at u, taking care that when one foot is placed at u, the other foot can touch ST in no other point but q. But, as the ray is now passing out of glass into air, o p is the sine of the angle of incidence, and Q U the sine of the angle of refraction; hence the line su drawn through u will be the refracted ray. The refraction of the prism has therefore bent the ray HR, which would have gone on to v in the straight line H R V, into the line s u, which forms with HV the angle u w v, which is the deviation or change of direction of the ray; so that if the ray HR proceeded from the sun, or other luminous body, it would, by an eye placed at u, be seen at x, in the direction uwx, and the angle of deviation will be H wx, equal to u w v.

In the preceding case the refracted ray RS, in passing through the prism, is parallel to its base BC; and, this being the case, the angle of deviation н wx is less than in any other position of RS, and therefore of н o, as may be readily proved by constructing the figure for any other position of these rays. If the eye be placed behind the prism at u, and the prism turned round, we shall find that RS is parallel to the base BC, by the image of the candle at a being stationary. When the prism is placed in the position that the ray Rs is parallel to в C, or perpendicular to a y, a line bisecting the refracting angle B A C of the prism, then it is evident that the angle of refraction at the first surface, b Rn, is equal to BAY, half of the angle of the prism. Now, as half this angle is known, and the angle of incidence HRM can be easily measured, we

have, without further trouble, the angle of incidence and the corresponding angle of refraction at the surface A B.

By the following proportion we obtain the refractive power-As the sine of the angle of refraction is to the sine of the angle of incidence, so is unity to the index of refraction—that is, dividing the sine of the angle of incidence by the sine of the angle of refraction we find the index. This is probably the simplest method, and the most generally applicable for measuring refractive powers or indices, because soft solids and fluids can be placed in the refracting angles of hollow prisms made by joining two plates of parallel glass.

Refraction of Light through Plane Glasses.—Let a b (Fig. 5) be a plane glass, and CD a ray of light refracted at D, on entering the glass in the direction D E, and at E, on going out of the glass, in the direction EF: if the direction of the refracted rays DE and EF be determined by the method shown at Fig. 2, it will be found that EF is parallel to CD; for, however much C D is bent out of its direction at the first surface of the glass, it is bent just as much in the opposite direction at the second surface, and will appear to an eye placed at F, as if it came in the direction G E, which is parallel

D

Fig. 5.

to CD. If we suppose any number of to fall upon

rays

the upper surface of the glass A B, in a direction parallel to CD, they will suffer the same refraction as DE, and pass out at the lower side in a direction parallel to EF. Hence parallel rays falling on a plane

glass will retain their parallelism after passing through

it.

If from any point c (Fig. 6) diverging rays, such as CD, CE, be incident upon a plane glass A B, they will be refracted into the directions DF, EG, by the first surface, and in the directions FH, GI, by the second. By continuing FD and GE backwards in the same straight lines, they will be

found to meet at J, a point farther from the glass than c. If we suppose the surface ED to be that of standing water placed horizontally, an eye within it would see the point

B

Fig. 6.

H

c removed to J, the divergency of the rays DF, EG having been diminished by refraction at the surface E D. But when the rays DF, EG, undergo a second refraction, as in the case of plane glass, we shall find by continuing HF, IG backwards in the same straight lines, that they will meet at a point к, and the object at c will appear to be brought nearer to the glass; the rays F H, G I, by which it is seen, having been rendered more divergent by the two refractions. A plane glass, therefore, diminishes the distance from it of the divergent point of diverging rays.

If we suppose the rays H F, GI to be converging to K, they will be made to converge to c by the refraction of the two surfaces, and consequently a plane glass causes the convergent point of converging rays to recede from it.

If the two surfaces DE, F G are equally curved, the one being convex and the other concave, like a watchglass, they will act upon light nearly as a plane glass, and precisely like a plane glass if the convex and