Telescopes and Spectroscopes for Lines of Finite Width. 323 



all of the elements of a line of uniform brightness will there- 

 fore be 



,=of 



SHI 2 J (?-<£) 



!>-*>]' 



(9) 



= C'\ sm X 



dx=f(y,')- ■ (io) 



The value of the definite integral (10) cannot be found 

 directly in terms of y and <r, but it can easily be evaluated by 

 mechanical quadrature for different values of these variables. 

 For a=a, which is about as small a value as is ever used in 

 practice, the values of Y=f(y) are given in Table I. For the 

 sake of comparison the values of I (from 8) are also included. 



Table I. 



7- 



r=/(r). 



I. 



r- 



r=/(y). 



I. 



•o 



100 



1-00 



•8 



•24 



•055 



•2 



•92 



•87 



10 



•11 



000 



•4 



•71 



•57 



1-2 



•044 



•024 



•6 



•45 



•25 



1-5 



•030 



•045 



The diffraction-curves represented by these values are 

 plotted in fig. 1 (p. 324). Like I, the curve V does not fall off 

 regularly, but passes through a series of maxima and minima 

 whose position is given by the general equation* 



iry 



y tan — - = tc tan^— for 2m/-/2m + l, 

 a 2 2a x « x 



2m. 



1 Try 2, 7T0-p _ 1/°" 



-tan— - = — tan -r- for 2m — 1< - 

 y u ar Zee N a. 



\xv \ 



* This part of the problem, i. e., that of locating the position of the 

 minima in the diffraction-patterns of a slit and of a circular aperture of 

 finite width, was worked out by the writer (at the suggestion of Professor 

 Michelson) about six years ago, while a student at Clark University. 

 The results were published in Professor Michelson's paper on u Applica- 

 tion of Interference Methods to Astronomical Measurements " (Phil. 

 Mag. July 1 890, p. I, see pp. 14-17). 



