800 Transactions of the Society. 



Having already settled the total amplification N, wbich is 

 required for tlie utilization of a given aperture,* we need only to 

 find tlie super-amplification v wbicli an objective of tbat aperture 

 will bear, without a perceptible depreciation in the quality of the 

 image if objectives up to the present standard of excellence are 



supposed. The quotient — will then indicate at once the normal 



amplification [N] of the objective which is necessary in order to 

 obtain the said N under the best possible conditions ; and having 



thus determined [N], the quotient r^. will yield the focal length 



which an objective of the aperture in question ought to have for 

 utilizing the delineating-power of that aperture in the most 

 favourable manner. (The focal length thus assigned for a given 

 aperture will be expressed by millimetres or by inches, according 

 as I is taken = 250 mm. or = 10 inches.) 



Though the problem in this way leads us to practical questions 

 which are to be answered by observation, apart from all theory, it 

 will not be useless to point out some theoretical considerations 

 which may elucidate certain experimental facts, or guide the 

 observer in experiments on that subject. 



(a) One fact which we may foresee in theory, is that the limit 

 of useful super-amplification must depend on the aperture of the 

 objectives, and that the former must diminish with increase in the 

 latter. The greater the aperture the wider the range for the 

 deviation of the rays from the ideal collection of the pencils to 

 mathematical image-points. All technical faults of the lenses — 

 slight defects of figure and of centering — must give rise to increased 

 deviations, and therefore to an increased amount of their accumu- 

 lated efiects, because the clear diameter of the lenses which transmit 

 the pencils bears a greater ratio to those radii of curvature which 

 are required for the wider aperture. Exactly the same holds good 

 with the strictly residuary aberrations which are the predominant 

 source of defective performance in modern objectives (the un- 

 avoidable technical faults being much less apparent with the 

 excellence of workmanship which has now been attained). 



The sources of the residuary aberrations in question are perfectly 

 well known in theory. Some of them result from the dispro- 

 portionate increase of the positive and negative spherical aberra- 

 tions in difierent parts of tlie system, arising from the increase of 

 obliquity and in regard to difierent colours, which disproportionality 

 prevents a strict compensation of the opposite spherical aberrations 

 even for the rays of one colour and gives rise to still more con- 

 siderable residuals in the totality of the rays of mixed light. 



* Sec tliis Journal, ii. (1882) p. 463. 



