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The integrals in (6) may be at once expressed in terms of the so-callei sine-integral and cosine-integral denned by
a. , .      f* sin as ,        n. ; .     fx<x>sas , Si (as) =      - dx,      Ci (x) -      - das. .'o    *                         Joo   x
If the limits of % be fx and |-2 we get
sin T[Si {(0 + <) & - Si 1(0 + ft &} + Si {(0 - 0) } - Si {(0 - 0) }J + cos T[Ci {(0 - 0) &} - Ci {(<? - <) I,} - Ci {(0 + </>) ,} + Ci {(6 +
If _ =  2 =  f , so that the second aperture is symmetrical with respec to the axisa the Ci's, being even functions, disappear, and we have simply
2 sin T [Si ((# + <)} + Si {(0 -(/>)}] ................ (8)
If the aperture of the telescope be not purposely limited, the value of  or rather of K, is very great, and for most purposes the error will be smal in supposing it infinite. Now Si (+ oo )= + |TT, so that if < is numerically less than 0, I = 47T2, but if < is numerically greater than 6, 1=0. Th angular field of view 2$ is thus uniformly illuminated and the transition tc darkness at angles  0 is sudden  that is, the edges are seen with infinite sharpness. Of course, f cannot really be infinite, nor consequently the resolving power of the telescope ; but we may say that the edges are defined with full sharpness. The question here is the same as that formerly raised under the title "An Optical Paradox*," the paradox consisting in the full definition of the edges of the first aperture, although nearly the whole of the light at the second aperture is concentrated in a very narrow band, which might appear to preclude more than a very feeble resolving power.
It may be well at this stage to examine more closely 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
J can be expressed by means of the Si-funcfcion.    As may be verified by differentiation,
7?-1sm27;,   ...................... ...(10)
vanishing when 17 = 0.   The places of zero illumination are defined by 77 = 'HTT, when n = I, 2, 3, &c. ; and, if 77 assume one of these values, we have simply
Si(2n7r) ........ ................. (11)
Phil. Mag. Vol. ix. p. 77,9 (1905); Scientific Papers, Vol. v. p. 254.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,