Retardations, and on the Theory of Foucault* s Test. 167 



The integrals in (6) may be at once expressed in terms of 

 the so-called sine-integral and cosine-integral defined by 



J" sin x i "''cos r 

 dx, Ci(V)= 1 -dx. 

 x •'. * 



If the limits of f be f x and f 2 we get 



dnT[Si{(0+*)&l-Si{(tf+#)&} 



4- Si{(0-£)f 2 }-Si{ (<?-£)&}] 



+ C osT[Ci{(0-<£)f 2 }-Ci{(0-<£)f 1 } 



_Ci{(0+<£)f 2 } + Ci!(0 + </>)? 1 }. . (7) 



If f i = — f 2 = ~^ ? so that the second aperture is symmetrical 

 with respect to the axis, the Oi's, being even functions, dis- 

 appear, and we have simply 



2 s inT[Si{(0 + £)?} + Si{(0-</>)»3 . . . (8) 



If the aperture of the telescope be not purposely limited, 

 the value of f, or rather of /ef, is very great, and for most 

 purposes the error will be small in supposing it infinite. 

 Now Si(±<*>) = ±^7r, so that if <f> is numerically less than 

 6, I = 47r 2 , but if (f> is numerically greater than 6, 1 = 0. The 

 angular field of view 20 is thus uniformly illuminated and the 

 transition to darkness at angles +6 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 narrowband, 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 of light at £ is 

 proportional to f ~ 2 sin 2 6% or, if we write 77 for 6%, to 

 n~ 2 sin 2 7}. The whole light between and n is thus 



* Phil. Mag. vol. ix. p. 779 (1905) ; ' Scientific Papers,' vol. v. 

 p. 254. 



