166 Lord Rayleigh on Methods for detecting small Optical 

 point f of the second aperture will be represented by 



or, if xlf=d, by 



jj dO sin K (Yt-f-B, + 0%), (3) 



the limits for 6 corresponding to the angular aperture of the 

 lens A. For shortness we shall omit k*, which can always 

 be restored on considering " dimensions/' and shall further 

 suppose that R is at most a linear function of 6, say p + crd, 

 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 

 Yt—f, and if T be written for Yt—f—p, (3) becomes 



sinTJd0cos(f~<r)0 + cosT$d0sin(f-o-)0. /., (4) 



Since the same values of p, a apply over the whole aperture,, 

 the range of integration is between +#, where denotes the 

 angular semi-aperture, and then the second term, involving- 

 cos T, disappears, while the effect of a is represented by a 

 shift in the origin of f, as was to be expected. There is now 

 no real loss of generality in omitting R altogether, so that (4) 

 becomes simply 



2ginT sinf0 ? 5 



? 



as in the usual theory. The borders of the central band 

 correspond to £0, or rather a:£(9, = +7T, or £0= ±^\ 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 



J S J f 



»«og T j' cos (*-0)g--™» <* + »% . . (6) 



It will be observed that, whatever may be the limits for £ r 

 the first integral is an even and the second an odd function 

 of cf>, so that the intensity (I), 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 h = 2ir. 



