whenever t/s is less than 031. If the second term cannot be neglected it is easily computed with sufficient accuracy.

If the spherometer were accurately made, the centre of the screw would move along a line through the centre C of the circumscribing circle A,A,A,, perpendicular to the plane А12А. If it does not pass through C but through a point C' distant a', b', c' from the point  ̧ ̧, then the distance CC' (= d) is given approximately by the equation

1 2 31

• d2 = ‡ {2 (r− a2 + r—b22 + r− c'2) + r− a' r — b′

+ r− a' r — b' + r — b'r-c')}.

Now consider a section of the surface by a plane through O and the centre of curvature of the surface C'C, i.e. the circle ANDN' (Fig. 65). The plane

through the points A, A, A, is represented

by the line ACD.

2 3

Assuming that when the instrument is placed on a horizontal surface the axis. of the screw is vertical, the distance h through which the point of the screw has to be moved when the instrument is placed on the spherical surface, will be C'P.

PN

A

C' C

P'N'

Fig 65.

Now C'P.C'P' = C'A. C'D and if ON=R, PP'2/R2-d' and C'P'-2√R2-d2-h. h (2√ R2 — d2 − h) = (r — d) (r + d),

Hence

or if d is small,

=

-√ (p2 + h2 − d2)2 + 4d2h2,

R = (r2 + h2)/2h.

Calculate R from one of these equations, and determine similarly the radius of curvature of the other surface of the lens. Verify by the following optical method the values found: I. For a Concave surface.

Place a screen provided with the hole and cross wires at one end of the optic bench with the white side towards the centre of the bench. Place a bright flame behind the screen.

Place the lens on the bench with its axis parallel to the axis of the bench, and the surface the curvature of which is to be measured towards the screen. Find the position of the surface when the image of the cross wires on the screen produced by reflection at the surface is most distinct. The cross wires are then at the centre of curvature of the surface.

II. If the surface is convex the radius of curvature may be determined optically by the following method :

C (Fig. 66) is the lens the radii of curvature of which we have to determine. B is an auxiliary lens provided with a stop

C

Fig. 66.

of small diameter. A is the screen in the centre of which is a small hole with a cross wire, which is illuminated as shewn. Keeping the distance AB, which must be greater than the focal length of B, constant, we can find a position of C such that the rays of light, after passing through B, strike the surface of C perpendicularly, and returning along the same path, form an image of the cross wire on A. Take the reading of the position of C, then remove C, and place a screen D in such a position that a distinct image of the cross wire is obtained on it, the lens B being kept in the same position. The distance from the lens to D is then the radius of curvature of the face nearer the cross wire. By turning the lens C round, the radius of curvature of the other face is similarly obtained. The apparatus used is shewn in Fig. 67.

Fig. 67.

Determine approximately the focal length of the lens, if convex, by placing a screen behind it at such a distance that an inverted image of some distant object is formed on the screen. The distance of the screen from the nearest point on the surface of the lens is the focal length approximately.

To determine it more accurately, if the lens is a converging one, mount it with a plane mirror immediately behind it. Place the mounted lens and the mirror on the optic bench.

Behind the hole and cross wire place a luminous burner, and find the position of the lens and mirror for which a distinct image of the cross wire is projected on to the screen, in the same plane and as nearly as possible coincident with the cross wires themselves (Fig. 68).

The light from F in the direction FA is refracted by the lens and proceeds after refraction in the direction AD. If AD is perpendicular to the surface of the mirror, the reflected ray traverses the same path as the incident, and is, therefore, brought to a focus again at F.

In order to determine the distance of the surface of the lens from the screen read the position of some part of the lens-stand on the scale of the optic bench used. Move the lens towards the screen till it touches one end of a rod of known length the other end of which touches the cross wire. Again read the position of the lens-stand. The sum of the length of the rod and the distance through which the stand has been moved is the distance of the surface of the lens from the cross wire. The focal length may be found from this by adding to it the thickness of lens.

If the lens is a divergent one, place between it and the cross wires a convergent lens. Place behind the two and close to

the divergent lens, a plane mirror (Fig. 69). Move the divergent lens and plane mirror till a distinct image of the cross wires is formed on the screen close to the wires.

Note the position of the divergent lens, remove it and the mirror and place behind the converging lens a screen in such a position

A

Fig 69.

that an image of the cross wires is formed on it. The distance from the screen to the position previously occupied by the divergent lens is the focal length required.

The focal length of a lens is connected with the radii of curvature of its surfaces and with the index of refraction μ of its material by the equation

where R1 and R2 are taken positive for convex and negative for concave surfaces.

Calculate μ from the observations, and tabulate as follows:

SECTION XXXVII.

DETERMINATION OF THE INDEX OF REFRACTION OF A LIQUID BY TOTAL REFLECTION.

Apparatus required: Horizontal graduated scale with upright and slit, ebonite block, and glass cube.

When a ray of light traversing an optically dense medium impinges on the surface of separation of that medium from a rarer medium, making an angle with the normal at the point of incidence greater than the "Critical Angle," the ray is totally reflected, no part of it entering the rarer medium. If N is the index of refraction of the denser, n that of the rarer medium, the least value of the angle of incidence for which total reflection can take place, i.e., the Critical Angle, is given by the equation

sin 0 =n/N.

A glass cube of about 4 cms. edge is provided. On one face a line parallel to and about a cm. from an edge has been drawn with a diamond.

Put a few drops of water on the ebonite block provided, and place on it the cube with the face on which the line is drawn vertical, the line itself horizontal, and a cm. above the ebonite. A film of water will be formed in immediate contact with the lower face of the cube. Now place the ebonite and cube on the scale and about 30 cms. behind the slit in the upright. Place the scale on a table in front of a window through which the sky can be seen. Look through the slit at the lower surface of the cube, and notice that on moving the cube and block towards the slit, the appearance of this surface changes from bright to