SPHERICAL ABERRATION. 



61 



correspond to the object-distance *5. The focal lengths of the 

 marginal and central rays will then be expressed as follows : 



The spherical aberration appears to be completely eliminated; 

 but this holds good only for marginal rays of 30, and does 

 not prove the combination of lenses to be aplanatic to the 

 greatest extent for the other zones of rays, nor, therefore, for 

 the whole effective cone. The inner portion of the cone of 

 rays might possibly give a more favourable image if the distances 

 of the lenses were varied. The working out of the calculation 

 for all the inner rays to every 5 or 10 would determine this 

 point. 



Voluminous as these calculations are for each different case, 

 until at length data are found which give a minimum of aberration, 

 they should unquestionably be undertaken if the problem could 

 thus be solved once for all, of producing objectives of the greatest 

 possible perfection in any desired quantity. But it is one of the 

 most difficult problems of the optician's art to satisfy even 

 approximately the demands of the mathematician in the con- 

 struction of lenses. In most manufactories 1 the unavoidable in- 

 accuracies of workmanship are so great that the calculations, in 

 fact, retain a practical value only for general guidance, and all 

 beyond is left to the skill and experience of the workman. 



As already pointed out, in the construction of the double-lenses, 

 when the optician wishes to produce either a slight degree of 

 under- or over-correction, or perfect aplanatism, he must select 

 those pairs of lenses whose combined action approximates to 

 that of his standard lenses. The proper combination of the 

 double-lenses to form the objective demands care and judgment. 

 Many modern objectives are built up each complete in itself, so 

 that for different amplifications (with the same tube and eye- 



1 In the " Archiv fiir mikr. Anat." Bd. ix. p. 414, Abbe states that for some 

 time past C. Zeiss, of Jena, has been manufacturing both high and low powers, 

 equal to any produced hitherto, strictly according to calculated formula). 



