136 Prof. F. L. 0. Wadsworth on the 



In dealing with a general problem of this nature, there 

 are always three expedients that may be adopted with a 

 greater or less degree of success : — 1st, altering the focus of 

 the instrument ; 2nd, diaphragming the optical surface of the 

 instrument or limiting the field ; 3rd, altering the form of 

 the reflected wave-front, generally by altering the form of 

 the optical surface itself. Let us take the two most favour- 

 able types of O.S. mounting, the A-B and the F forms, and 

 see what can be accomplished in the way of improving the 

 definition by any one or a combination of these means. 



1st. Effect of alterations of focus. By examining the 

 Table we find that in every type of O.S. mounting save one, 

 A, the numerical values of Zi and Zo, the aberrations of the 

 two halves of the grating-surface, are different. Since these 

 aberrations (with the single exception noted), are in large 

 part due to the unsymmetrical action of the lateral elements 

 Z>, V, fig. 1, and are of opposite sign, it follows that a slight 

 improvement in definition may be secured by shifting the 

 focal point, 0^, by a small quantity du such that 



f?M = J [longitudinal aberration due to Zi 



—longitudinal aberration due to Z2] ... (31) 



1 W 



j^gr /-v 1 ^'' (p sm yS cos ^/ 



-0-002KZi-Z2) ""^P— (31a) 



— '^^ ^ cos 6/+ cos 2 ' ^ ^ 



in which the values of Zi and Zg are to be taken without 

 regard to sign. In most cases the numerical differences, 

 Zi — Z2, are so small compared with Z(max.) that no very great 

 improvement can be effected by departing from the focal 

 curve defined by (6). In any case the departures will be 

 very small. Thus in case A-B the numerical differences 

 Zi — Zg at the edges of the field are always less than 0*04. 

 Hence we have for the maximum values of du (f = 60°). 



^^*max.;ro-oooo5p 



or, for a Rowland grating of the usual radius of curvature, 



^Wmax.= 3 mm. 



Although no great improvement can be effected by a 

 * Astrophysical Journal; vol. xvii. p. 1, January 1903. 



