124 Prof. F. L. 0. Wadsvvorth on the 



which is the same as the expression for Zi, with the exception 

 of the sign of the term 0. 



From the general theory of the grating* \\i\ always have 

 for any given spectral image 



p sin /3 (sin d — sin i) = mn\. 



Also for parallel incident lightf 



2^(cos ^+ cosi)— pcos^^ = (6) 



The first two terms of equations (4) and (5) are therefore 

 always satisfied when the spectrum is in focus. The remain- 

 ing terms express the aberration at the primary focal point o' 

 for any given values of d, i, and p. 



From (6) we have for it 



pcos^^ _, 



y,z=z .. ...... (7) 



cos'^ + cosi* ••'••• V / 



Substituting this value of u in the terms 0, P, and Q of 

 equations (4) and (5) and reducing, we obtain as the general 

 expression for the primary aberration of a concave grating 

 used with parallel incident light 



ry P ' 'iO ■ n .(cOSi + COS^) 



Li= — '- sm''n sni 6 cos i ^7^ 



I C0S*'t7 



+ |sin*/3^^^^^^i^^' [cos^z-(cos6> + 2cosz)nang26>]. (8) 



b COS^^ ^ ^ / o J' V / 



and for Zg the same expression, save for the opposite sign of 

 the first term in sin^/^. The corresponding general expres- 

 sion for the primary aberration of a concave grating mounted 

 in the usual Rowland manner, as deduced by Glazebrook and 

 Rayleigh, is 



Zo= [ii + 1^' = central ray) — (lateral ray) J 



= — ^/o sin'^/S (sin d tang -\- -in i tang i) . . (9) 



Comparing (8) and (9) we see that in general the effect of 

 aberration is not only larger, but is of a more prejudicial 

 character, when the grating is used as an objective spectro- 

 scope than when it is used as originally proposed by Rowland. 

 In the former case the aberration is unsymmetrical on account 

 of the term in sin^^ ; in the latter case this term is eliminated 



* See, for example, Eowlnnd, Phil. Mag. vol. xvi. (1883). 



t See Astropliysical Jouraal, vol. ill. p. 00. 



\ In Glazebrook and Rayieigli's papers the aberration is expressed as 

 the difference, lateral ray— central ray, and is therefore of the opposite 

 sio-n. 



