If
1917]
AND ON THE THEOKY OF FOUCAULT's TEST
457
the same year, and in subsequent publications* he made many interesting applications, such as to sonorous waves in air originating in electric sparks, and further developed the technique. His most important improvements were perhaps the introduction of a larger source of light bounded by a straight edge parallel to that of the screen at the observing end, and of a small telescope to assist the eye. Worthy of notice is a recent application by B. Cheshire j- to determine with considerable precision for practical purposes the refractive index of irregular glass fragments. When the fragment is surrounded by liquid j of slightly different index contained in a suitable tank, it appears luminous as an irregularity, but by adjusting the composition of the liquid it may be made to disappear. The indices are then equal, and that of the liquid may be determined by more usual methods.
We have seen that according to geometrical optics (\ = 0) the regular light from an infinitely fine slit may be cut off suddenly, and that an irregularity will become apparent in full brightness however little (in the right direction) it may deflect the proper course of the rays. In considering the limits of sensibility we must remember that with a finite \ the image of the slit cannot be infinitely narrow, but constitutes a diffraction pattern of finite size. If we suppose the aperture bounding the field of view to be rectangular, we may take the problem to be in two dimensions, and the image consists of a central band of varying brightness bounded by dark edges and accompanied laterally by successions of bands of diminishing brightness. A screen whose edge is at the geometrical focus can cut off only half the light and, even if the lateral bands could be neglected altogether, it must be further advanced through half the width of the central band before the field can become dark. The width of the central band depends upon the horizontal aperture a (measured perpendicularly to the slit supposed vertical), the distance / between the lens and the screen, and the wave-length X. By elementary diffraction theory the first darkness occurs when the difference of retardations of the various secondary rays issuing from the aperture ranges over one complete wave-length, i.e. when the projection of the aperture on the central secondary ray is equal to X. The half-width (f) of the central band is therefore expressed by f =/X/a.
If a prism of relative index /z, and of small angle i, be interposed near the lens, the geometrical focus of rays passing through the prism will be displaced through a distance (p, - 1) */. If we identify this with £ as expressed above,
we have
(p-l)i = \la, ..............................(1)
* Pogg. Ann. Bd. oxxvm. p. 126 (1866); Bd. oxxxi. pp. 33, 180 (1867).
f Phil. Mag. Vol. xxxn. p. 409 (1916).
t The liquid employed was a solution of mercuric iodide, and is spoken of as Thoulefc's solution. Liveing (Camb. Phil. Proc, Vol. in. p. 258, 1879), who made determinations of the dispersive power, refers to Sonstadt (Ghem. News, Vol. xxix. p. 128, 1874). I do not know the date of Thoulet's use of the solution, but suspect that it was subsequent to Sonstadt's.
f I
Ioung's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.