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



of a wave-period may pass unnoticed. On this point Lord 

 Rayleioh * very pertinently remarks : — " The best results will 

 be obtained with an aperture . . . giving an extreme aberra- 

 tion of from a quarter to half a period . . . with an increased 

 aperture/' (with proportionally increased aberration), " aber- 

 ration is not so much a direct cause of deterioration as an 

 obstacle to the attainment of that imj^roved definition ivhich 

 shoidd accompany the increase of aperture.'''' In other words, 

 there will be no discrepancy between Mitchell's standard 

 of " excellent definition '^ over a 7" field and the conditions 

 of equation (37), if the resolving-power of his instrument did 

 not exceed 210 units \, On the same basis of comparison, 

 the definition would be equally " excellent '^ with an instru- 

 ment of the resolving-power actually used (about 85,000), 

 although the aberration would be enormously increased. In 

 this connexion it should be noted that the character of the 

 spectra photographed was such that it would be natural to 

 apply a less rigorous standard of definition to the results than 

 would be applied, for example, to solar-spectrum plates. As 

 the writer has previously shown J, a resolving-power of not 

 more than 2000 is required to obtain as good definition on 

 bright-line hydrogen-spectra as is obtained with a power of 

 400,000 on fine-line absorption-spectra from the sun. 



Fortunately there is a much more definite and impersonal 

 standard of performance of an optical instrument than that 

 of definition. This is the standard of accuracy or metro- 

 logical power, which has recently been investigated in some 

 detail by the writer §. As has been there shown, the limiting 

 metrological power e of an instrument is about one-fifteenth 

 as great as the limit of resolving-power a, or the limit of 

 accuracy A is 



A — 15r. 



For the grating used by Mitchell the value of A is about 

 1,270,000, and the limit of metrological power e is therefore 

 about 0'004 tenth-metre. This should be the mean or 

 " brute ^' error (not the probable error) of an individual 

 setting on a spectrum-line, with a grating of this resolving- 

 power used under the best optical conditions. 



* '' Influence of Aberration," Pliil. Mag. vol. viii. p. 403, Nov. 1879. 



t Obtained by puttino- k=7° and (3= -073 in (69), and solving for r. 



X Phil. Mag. vol. xliih p. 336. 



§ " On the Optical Conditions required to secure Maximum Accuracy 

 of Measurement in the Use of the Telescope and Spectroscope,"' Astros- 

 physical Journal, vol. xvi. p. 267 ; vol. xvii. pp. 1 and 100, Dec. 1902 

 and Jan.-Mar. 1903. 



