Telescopes and Spectroscopes for Lines of Finite Width. 327 



These results are plotted (crosses) in fig. 3. The agree- 

 ment with the first part of the curve obtained by theory is 

 very good, but beyond the point <r = a the experimental values 

 are considerably less than the theoretical ones. These last 

 were obtained, it will be remembered, on the assumption that 

 in order to obtain resolution the difference in intensity between 

 the centre and edges of the diffraction-pattern of a double 

 source must be at least 20 per cent. These results indicate 

 that when the lines are broad a falling off in intensity at the 

 centre of considerably less than this is noticeable. Indeed 

 this is what we should expect, since we know that on an 

 extended bright background (such as a planetary surface) 

 faint markings may be distinguished where the variation in 

 intensity from the background itself is not more than from 

 two to five per cent. 



We are therefore certainly on the safe side in following the 

 curve deduced from theory. The value of 2, the angular 

 resolution of the telescope for the wide lines, is, moreover, 

 practically the same whichever curve be followed, because, 

 for the values of a for which the two curves begin to depart 

 to any extent from each other, the value of 8 is small com- 

 pared to <7 itself*. 



The theoretical curve of fig. 3 may, up to the point <7=3<z, 

 be closely represented by the hyperbola of the form 



" + I = 1—, (11) 



whence we get 



* 2 



X 



*=27+-« <"» 



But a- °- and a= £ (for rectangular aperture m = l). Sub- 

 stituting these values we get 



a 1 ^ 2 



Bss b2+ + X' ■ ' • • • (13) 



The angular distance between two lines of width a which 

 can just be resolved is then 



2=<7+8= £0 + WVx x )- • • «.<") 



* For the value of a=l-5a, for which the difference between the two 

 curves is greatest, the two values of 2 differ by only about four per cent. 

 ±or a=3a the difference in 2 is only about two per cent. 



