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AND ON THE THEORY OF FOUCAULT'S TEST
a small angle <£. P is a point in the first aperture, AP = as, BQ = |, AB =/. Any additional linear retardation operative at P may be denoted by R, a function of as. Thus if V be the velocity of propagation and tc = 2-Tr/X, the vibration at the point £ of the second aperture will be represented by
or; if &//= 0, by
the limits for 6 corresponding to the angular aperture of the lens A. For shortness we shall omit «*, which can always be restored on considering "dimensions," and shall further suppose that R is at most a linear function of 6, say p + a-0, or, at any rate, that the whole aperture can be divided into parts for each of which R is a linear function. In the former case the constant part p may be associated with Vt —f, and if T be written for Vt —f— p, (8) becomes
Since the same values of p, or apply over the whole aperture, the range of integration is between + 0, where 0 denotes the angular semi-aperture, and then the second term, involving cos T, disappears, while the effect of or is represented by a shift in the origin of £, as was to be expected. There is now no real loss of generality in omitting R altogether, so that (4) becomes simply
_ . ™ 2 sin T —
as in the usual theory. The. borders of the central band correspond to £0, or rather «£#, = ± TT, or £0 = ± £X, which agrees with the formula used above, since 20 = a/f.
When we proceed to inquire what is to be observed at angle </> we have to consider the integral
d>)£ + sin(0-<p)£ ,
. + cos T
It will be observed that, whatever may be the limits for £, the first integral is an even and the second an odd function of (f>, so that the intensity (/), represented by the sum of the squares of the integrals, is an even function. The field of view is thus symmetrical with respect to the axis.
Equivalent to supposing \=2ir.a point in the focal plane of the telescope, or of the retina, may be regarded as a parallel pencil inclined to the axis at