Telescopes and Spectroscopes for Lines of Finite Width , 331 



in finite form. They may be integrated by developing into 

 a series, but I have found it easier and quicker to integrate 

 by mechanical quadrature. Owing to the very close corres- 

 pondence between the curves represented by (17) and (18) 

 (see fig. 4), the result will be practically the same whichever 

 law be adopted. The expression for I x is the one which has 

 actually been integrated, and the resulting curves yfr^/c, 7, «,) 

 for two values of k are given in fig. 5. The dotted lines 

 represent the curves /(<£) and the full lines the resulting 

 diffraction-pattern ty x (7) . 



Fig. 5. 



O 2.0 19 



For convenience the values of k are expressed in terms of 

 the " half-width " of the line (Michelson) and a. the limiting 

 resolving power of the spectroscope objective. The "half- 

 width" 3 is defined to be the value of <£ for which /</> = !. 

 Hence 



_ Nap. log 2 



(21) 



What we may call the effective width of the line iv is the 

 width ab (fig. 4), which is equal to 48. At the points a and 

 b the intensity f(<p) is only about one-twentieth the intensity 

 at the centre, and the part of the curve beyond this point 

 may therefore be considered as having but little effect either 

 on the eye or on the photographic plate. 



The values of w in the curves of fig. 5 are w = 2a, 10 = 4a. 



In fig. 6 a the diffraction-curve for a double source, of 

 which each component is of width w = 2<z, is shown. Adopt- 

 ing the same rule as before, t. e. that for resolution the 

 intensity at the middle of the diffraction-pattern must not be 



