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It will be seen that the direction of the discontinuity ($ = 0) is strongly marked by excess of brightness, and that especially when p is small there is a large variation with the sign of p.
Perhaps the next case in order of simplicity of a variable R is to suppose R = 0 from 6 = - 0 to 6  0, and R = crd from 6 = 0 to 6 = -f 0, corresponding to the introduction of a prism of small angle, whose edge divides equally the field of view. For the vibration in fche focal plane we get
sin ( - o-) (91  , _ ,r fl - cos ( - a-) d    1 - cos %6~
.    , sin T
m T
In order to find what would be seen in direction <, we should have next to write (T+(p%) for T and integrate again with respect to  between the appropriate limits. As to this there is no difficulty, but the expressions are rather long. It may suffice to notice that whatever the limits may be, no infinity enters at < = 0, in which case we have merely to integrate (36) as it stands. For although the denominators become zero when f = 0 or  := cr, the four fractions themselves always remain finite. The line of transition between the two halves of the field is not so marked as when there was an actual discontinuity in the retardation itself.
In connection with these calculations I have made for my own satisfaction a few observations, mainly to examine the enhanced brightness at the edges of the field of view. The luminous border is shown in Draper's drawing, and is described by Topler as due to diffraction. The slit and focussing lens were those of an ordinary spectroscope, the slit being drawn back from the " colli-mating " lens. * The telescope was from the same instrument, now mounted independently at a distance so as to receive an image of the slit, and itself focussed upon the first lens. The rectangular aperture at the first lens was originally cut out of the black card. The principal dimensions have already been given. A flat paraffin-flame afforded sufficient illumination. The screens used in front of the telescope were razor-blades (Gillettes), and were adjusted in position with the aid of an eyepiece, the telescope being temporarily removed. It is not pretended that the arrangements used corresponded fully to the suppositions of theory.
The brightness of the vertical edge of the field of view is very conspicuous when the light is partly cut off by the advancing screen. A question may arise as to how much of it may be due to light ordinarily reflected at the edges of the first aperture. With the aperture cut in cardboard, I think this part was appreciable, but the substitution of a razor-edge at the first aperture made no important difference. The strongly illuminated border must often have been seen in repetitions of Foucault's experiment, but I am not aware that it has been explained.xn. p. 409 (1916).