142 Prof. F. L. O. Wadsworth on the 



From (33) with the .same notation above used we get 



4a^ tang «„, + a sin ^(1 - tang -fc,n) = '^ \y ^[^-^^q ' 



whence 



. 1 / \ asin/3\r. , lab 



VWsin«/3 2 ;]•••• ^ ^ 



The usual Rowland grating is of from 70 to 75 millimetres 

 semi-aperture and 650 to 660 centimetres radius of curvature. 

 For such a grating /3^/5i = '01l4- , and we find for k^ 



.2rO°-8— 50'for /=15^ 



«+,„ = 2°20' 



= ±0 \ 



As the value of ,8 increases the available field At^,„ of good 

 definition decreases and becomes zero, i. e. is confined to the 

 central line of the field, at the point 



X, _a sin /? 



wsin^yS r~ 



which gives us the same condition as expressed in (38). 



The only way in which we can increase the value of k^ and 

 /3„i is to decrease W. The maximum permissible angular 

 aperture varies inversely as the cube root of the linear 

 aperture, while the maximum field varies very nearly inversely 

 as the first power of the linear aperture. The maximum 

 spectral field of good definition varies therefore, very nearly 

 as the inverse square of the angular aperture. 



3rd. Effect of changing form of difcacted wave-fronts 

 The results of the preceding investigations show that the 

 ordinary Eowland concave grating, whose ruled surface is a 

 portion of a sphere, has an extremely restricted field of good 

 definition when used as an objective spectroscope, and that 

 althouo-h this field mav be somewhat increased bv sho-ht 

 readjustments of foci and in some cases by diaphragming off 

 a portion of the ruled surface, the improvement resulting 

 from such changes is small. It remains now to determine 

 whether any better results can be secured by altering the 

 form of the ruled surface or the character of the ruling. To 

 do this we must first derive a general expression similar ta 

 that for Z, equation (4), which will express the relative 

 retardation of the extreme ray bo^ with reference to the 

 central ray ao' when the light is incident on a surface of 

 anv form. This mav be done in one of two wavs. 



