Substituting t1='531, we find w2 = —

12.6EI

(I1 — I2) l'

This determines the speed of a simple oscillatory vibration. For such a vibration to be possible it is necessary that 12 be I2 greater than I, or for a solid circular cylinder, that l1 is greater than 0·866d1.

The normal whirling speed in this case is determined from

u1 = mu1lt1 (1 — t1) [ {t, (2—t,) — 1,2} — ¿t1 (1—t1)(2—t,)2].

1

We thus have a simple normal whirling speed corresponding

EI

to w2=66•2 and a simple oscillatory vibration corre

ท

sponding to w2=12·6

EI
(I2 — I1) / *

The ratio of oscillatory speed to normal whirling speed is

ml2 5.25 (12-11)

§ 23. For a uniformly loaded uniform shaft supported in three bearings the general solution is well known. It is given by

cotyl+cotyl′ = coth fl+coth l

and particular solutions by law and

taneously, where a and b are integers.

the mass per unit length.

l'=bπ simul

mw2

*=

where m is

EI

corresponding

The following Table gives the values of

to the first and second whirling speeds, and singular solutions,

for different ratios of lengths of span.

LXXVI. On the Diffraction of Light by Cylinders and Spheres immersed in a Medium of nearly equal Refractive Index. By NIHAL KARAN SETHI, D.Sc., Assistant Professor of Physics in the Benares Hindu University*.

[Plate XX.]

1. Introduction.

HE subject of the present investigation was suggested by a study of the beautiful and interesting phenomena observed when a mixture of equal quantities of turpentine and glycerine in a flat-sided flask is shaken up and allowed gradually to separate. Some of the effects noticed have been mentioned by writers on Optics †, but the matter does not appear to have been fully investigated. In order to obtain the best possible results, it is necessary to pay attention to the temperature of the liquids. Ordinarily, glycerine has a lower refractive index than turpentine throughout the visible spectrum, and the effects observed are not very striking; but on heating up the mixture, the refractivity of the turpentine falls off more rapi lly and ultimately becomes less than that of the glycerine. Very lively effects are obtained in the intermediate stages, the colour of the regularly transmitted light and the general appearance of the mixture as seen by the light scattered by it changing with the temperature. The superficial layer of the collection of drops which settle down through the mixture exhibits some particularly beautiful effects, due apparently to the large vertical gradient of refractive index which exists in the layer. The main features of the phenomenon are analogous to those observed in Christiansen's experiment, and have been dealt with as such in a recent paper by the author. There are, however, certain special features in the present case depending on the more or less exactly spherical shape of the individual drops which appear to be in the nature of interference and diffraction effects. It is proposed in the present paper to deal with these latter effects more fully and to give a theoretical discussion.

* Communicated by Prof. C. V. Raman, M.A.

+ E. g. L. Wright's 'Light,' p. 176, and R. W. Wood's 'Physical Optics,' 2nd ed. p. 112.

"On Wave-Propagation in Heterogeneous Media and the Phenomena observed in Christiansen's Experiment.” Proceedings of the Indian Association for the Cultivation of Science, vol. vi. parts iii. & iv. (1920).

2. Some Experimental Results.

With the liquids moderately warm and the mixture regularly transmitting some part of the spectrum, the two edges of each individual drop are observed to diffract light through considerable angles and shine with different colours, e. g. one edge may be red and the other blue or violet. The whole mixture consequently presents a curious appearance, with some points red and some points blue. On further warming, the region of equality of refractive indices is shifted towards the blue end of the spectrum, and the blue edge gradually deepens into a violet colour and ultimately disappears altogether, while the red one becomes more and more yellowish. At a very high temperature, almost that of boiling water, this latter edge becomes very bright and almost perfectly white, with perhaps a yellowish tinge, and this gives to the whole liquid a peculiar sparkling appearance.

An individual drop of glycerine, however, shows still more interesting effects when suspended at ordinary room temperature from a narrow g'ass tube in a flat-sided cell containing some turpentine. When a distant point source of white light is viewed through the drop held close to the eye, the diffracted light shows a number of coloured rings round a more or less white and broad central patch. The colours most prominent in these rings are red and greenish blue, and to some extent yellow. The inner rings are narrower than those in the outer part of the halo, the outermost one being very broad and almost achromatic. The light which forms these rings appears to come from the convex side of the drop, and the eye has to be taken round the edge in order to see the complete rings. The effect could not be observed at higher temperatures, both on account of the drop not remaining easily suspended and the difficulty of keeping the turpentine hot in the parallel-sided cell.

To find the explanation of these effects, some observations were made with the somewhat analogous case of a glass cylinder immersed in a parallel-sided cell containing a mixture of carbon disulphide and benzene of which the refractive index could be varied. The cylinder and cell are held at a distance of about half a metre from a narrow slit from which issues monochromatic light of considerable intensity, obtained from a 3000 c.p. quartz-mercury lamp by using a green ray filter. Putting the eye very close to the vessel and viewing the source through the cylinder, one sees in general a long band of light extending to very large angles on either side and broken up into a number of fringes, the nature of which is entirely different in the two cases in

which the refractive index μ of glass is less or greater than μ', the refractive index of the liquid which surrounds it. It is hardly necessary to say that the cylinder should be optically good, showing no striæ of any kind, otherwise very complicated effects are produced which may entirely obscure the real phenomenon.

3. Case (a) μ<μ.

Fig. 1 in the Plate is the photograph of a typical case of the phenomenon observed when the refractive index of the liquid is higher than that of the cylinder. The main feature of this is a number of fringes very narrow at first and gradually increasing in width in directions further and further removed from that of the incident light. (The fringes on one side of the source only are shown in the photograph.) The visibility of the fringes is also not constant. They are almost invisible at small angles, but gradually improve with increasing obliquity of observation till they attain almost perfect visibility and then deteriorate once again. Another important point about these is the fact that they become narrower and more numerous, and extend to greater angles as the difference between and increases. It is very instructive to perform the experiment with a monochromator in which the wave-length of the light can be continuously altered by the mere turn of a screw. As the wave-length for which μ-'=0 is approached, the fringe system closes and ultimately disappears altogether.

up

A clue to the explanation of these fringes is obtained when the surface of the cylinder is viewed by means of a small magnifying lens. It becomes clear at once that the whole of the light which forms this system of fringes arises from near one or the other of the edges of the cylinder. At each of these edges, one can clearly distinguish two very sharp and bright luminous lines on the surface of the cylinder from which the light appears to emerge, and which evidently act as interfering sources giving us the observed system of fringes. These luminous lines appear to be well separated from each other when observed at a small angle to the direction of the incident rays, and one of them is very much brighter than the other. As the direction of observation becomes more and more oblique, not only does the distance between the lines become smaller and smaller until it vanishes, but the brighter one diminishes and the feebler one increases in intensity until they become equal at a certain stage, beyond which the latter becomes the brighter of the two.

It will be seen from what follows that these features are in agreement with the observed changes in the visibility of the interference fringes in different directions.

4. Theory.

The effects observed may be explained on elementary principles in the following way :—

Let AQC (fig. 1) represent the principal section of the cylinder, and let XA and YPQ be parallel rays falling on it. The ray XA will follow the path ABCD, and the ray YPQ

π

2

incident at an angle greater than the critical angle will be totally reflected along QRS parallel to CD, if the angle of incidence at Q is equal to -(-i), where i and are the angles of incidence and refraction at A. These parallel rays CD and QRS must interfere when they are brought to a focus by a lens (e. g. of the eye *). The difference of path between the two rays is evidently

where a is the radius of the cylinder and μ its refractive index with respect to the liquid. This vanishes at the critical angle when sinip. In Table I. are set forth the values of & for various angles of incidence i, in a case where μ=99 and the diameter of the cylinder was 5 cm., which may be taken to be 10a λ. The angles of deviation A=2(-i) are also shown.

* There will also be a third ray diffracted from the edge of the cylinder, but its effect is not appreciable except at small obliquities, and then it practically coincides with QRS.