Method VI. Arithmetical interpolation by means of inverse squares of wave-lengths or squared wave-numbers.

It has already been stated that equal distances on the scale correspond very nearly to equal differences in the inverse squares of the wave-lengths. Hence calling y1, y2 and y the inverse squares of the wave-lengths of the two reference lines and of the unknown line respectively, and replacing the values of λ by the corresponding values of y in the result given in Method IV. we have:

From y, the wave-length is found by calculating 1/√y.

When the reference lines are not very close together this is the method which should always be adopted when arithmetical interpolation is employed.

Exercise III. Map a spectrum on the normal scale. Using the curve obtained in Exercise II., make as accurate a drawing as you can of the calcium spectrum on a scale of wavelengths, representing the characteristic features of the spectrum by one of the two methods of mapping spectra described on pages 185 and 186.

Exercise IV. Using the measurements previously made, calculate, by Methods IV. and VI. above, the wave-length of the sodium line, having given the wave-lengths of the lithium and thallium lines.

Enter observations and results as follows:

Notice that Method IV. gives only an approximate value, but that Method VI. gives a fairly accurate result. This shews that the reference lines are too distant from the unknown line for the first method of direct interpolation to be applicable, and for very accurate work even the method of inverse squares should not be used for lines which are so wide apart as these.

The wave-lengths and wave-numbers are here given to six significant figures, as the more complete values may be useful for reference, but for the purposes of the above exercises it will be sufficient to use four figures. The wave-lengths are taken as far as possible from Rowland's maps; in the case of Potassium, Lithium, and Thallium the numbers are those given by Kayser and Runge.

SECTION XLI.

THE SPECTROMETER AND ITS ADJUSTMENTS.

Apparatus required: Spectrometer, with Gauss eye-piece, plane parallel glass on stand.

An instrument in which the deviation produced by the passage of a beam of parallel light through a prism or other apparatus can be measured is called a spectrometer. It consists of a collimator S, a telescope T, and a horizontal divided circle C (fig. 83). The telescope moves about a vertical axis passing through the centre of the circle. Attached to the telescope is an arm carrying two verniers V, V' opposite

S

Fig. 83.

each other, which move along the circular scale, so that the angle through which the telescope is turned may be accurately measured. Above the divided scale is a table B, on which prisms or gratings may be placed. In some instruments

* In Fig. 83 the circle and verniers are enclosed in a case, and are read through windows in the case.

this table is fixed, but it ought always to be capable of being turned about an axis coincident with the axis of rotation of the telescope. The way in which an angular displacement of the table is measured differs in different instruments. If the divided circle is fixed to the collimator, the table should carry a second set of verniers. But in many instruments the divided circle is attached to the table. In that case, care must be taken that between any two readings of the verniers either the table only, or the telescope only is moved. If both are displaced, the vernier readings will only shew the difference between the angular displacements of table and telescope, not the actual displacement of either.

It is difficult to secure that the axis of rotation passes accurately through the centre of the divided circle. The error thereby introduced, called "error of eccentricity," is eliminated by having two verniers opposite each other as described. The angle measured by means of one vernier exceeds the correct value by as much as the angle measured on the other falls short of it; so that the arithmetic mean of the two results gives the correct angle. For a proof of this see Note, p. 202.

Examine the scale and vernier, and determine the value of the smallest subdivision of the principal and vernier scales. If e.g. the circular scale is subdivided into 20 minutes of arc, the vernier will probably be divided into 20 parts, of which every fifth will be numbered, and each of the 20 may be subdivided into 2 or 3 parts, which will allow the measurements to be made to 30 or 20 seconds of arc. In entering an observation in your Note-book, write down separately the readings of the two scales. Thus if the angle to be read off were 47° 43′ 20′′, the observation would be entered as follows:

A wooden model of the vernier is placed in the laboratory, and the student should practise reading it until he is quite familiar with it.

Before proceeding with the exercises students should study

the construction of the instrument they are using, and refer to a more detailed description, which will be supplied with the instrument. Special care should be taken to be familiar with the object of the various screw heads, which serve either to clamp some part of the instrument, or to give that part a slow motion. If the telescope, for instance, is to be pointed in any direction, it is first moved by hand as nearly as possible into its right position. It is then "clamped" by the proper screw, and finally brought to the proper position by means of a “fine adjustment screw," which can, within certain limits, alter its direction. For the purpose of clamping, it is not necessary to use great force. If the screw is screwed up gently it will be sufficient. If force is used the instrument will be damaged.

Other moveable parts of the instrument will also in general be provided with a clamping arrangement and fine adjustment. Before moving any part of the instrument care must be taken that the clamping screw of that part is released. All parts should move easily, and no force should ever be used.

In order to fix the direction in which the telescope points, a mark is placed in the focal plane of the object lens. This mark consists generally of a cross formed by two cocoon fibres, spider's threads, or very thin platinum wires. This arrangement, called the cross wires," can be turned in its own plane, so that one of the wires may be placed vertically, or they may both be placed at an equal inclination to the vertical. The latter position is most convenient when the telescope is to be pointed towards the image of a slit, for it is found easier to

place the centre of the cross on the image when the cross has the position shewn in Fig. 84, where AB represents the image