1917] AND ON THE THEORY OF FOUCAULT's TEST
Thus, setting n = 1, we find for half the light in the central band
461
On the same scale half the whole light is Si (oo ), or \TT, so that the fraction of the whole light to be found in the central band is
^ .....................
v '
lor more than nine-tenths. About half the remainder is accounted for by the light in the two lateral bands immediately adjacent (on the two sides) to the central band.
We are now in a. position to calculate the appearance of the field when the second aperture is actually limited by screens, so as to allow only the passage of the central band of the diffraction pattern. For this purpose we have merely to suppose in (8) that <?f =TT. The intensity at angle <£ is thus
4 Si !
V (f
.(13)
The further calculation requires a knowledge of the function Si, and a little later we shall need the second function Ci. In ascending series
«• / x 1 *31 1 ^
.(15)
7 is Euler's constant '5772157, and the logarithm is to base e
These series are always convergent and are practically available when SB is moderate. When x is great, we may use the semi-convergent series
fl 1.2 1.2.3.4 1.2...6
7T
— sin a?
x*
1
x
1
cos as •{— — a?
1.2 1.2.3.4
1.2.3 1.2.3.4.5
OK
X
.(16)
1.2 1.2.3.4
«3 a?
1.2.3
1.2.3.4.5
Xs
}
Tables of the functions have been calculated by Glaisher*. For our present purpose it would have been more convenient had the argument been Tr.r, rather than x. Between cc = 5 and x= 15, the values of Si (#) are given for integers only, and interpolation is not effective. For this reason some
Phil. Trans. Vol. OLX. p. 367 (1870).osely what is actually the distribution of light between the central and lateral bands in the diffraction pattern formed at the plane of the second aperture. By (5) the intensity oi light at f is proportional to £~2 sin2 6% or, if we write r) for 6%, to ?7~2 sin2 77, The whole light between 0 and 97 is thus represented by