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IT. — Discussion of the Formula of an hnmersion Objective of 

 greater Aperture than eorresi^onds to the Maximum possible 

 for Dry Objectives. By Mr. K. Keith. 



Plate LXXIII. (Lower iwition). 



De. Woodward having received from Mr. Tolles, and placed in my 

 hands, the elements necessary for the computation of the angular 

 aperture of the jo-inch objective, described by him in the number 

 of this Journal for November, 1873, p. 214, I have made the com- 

 putation with five figure logarithms. A computation of this kind 

 is much more satisfactory to mathematicians and more easily re- 

 viewed than an enlarged drawing. 



The objective is composed of seven lenses. The first four, com- 

 mencing at the back, are united by balsam into one combination, the 

 next two into a middle combination, and the seventh is a hemi- 

 sphere of crown glass. I give below the elements in a tabular form, 

 and annex a figure accurately drawn to a scale. Where distances 

 are given, the unit is in all cases an inch. 



The distance between the first combination and the middle one 

 is "008, and the radiant is assumed 10, from the first surface. 



From these elements I find that the extreme ray enters the first 

 combination " 09250 from the axis, at an angle with the axis of 

 0^ 31' 45"; and enters the middle 0* 07391 from the axis at an 

 angle of — 1 1° 1' 3", the negative sign indicating convergence of the 

 light. Adjusting the collar so that the distance of the front lens 

 from the middle is • 00528, I find that the same ray enters this 

 lens • 033 from the axis, at an angle of — 29° 44' 7", and makes, 

 after refraction, an angle of — 55° 17' 35" with the axis. 



Thus the extreme aperture, in fluid balsam, no allowance being 

 made for the setting of the small front lens, is 110° 35' 10". By 

 comjjutation the spherical aberration for this position of the collar 

 is practically nothing. If an allowance of only 0*00 162 be made 

 for the setting of the front lens, the aperture is reduced to 87°, and 

 this is doubtless a proper allowance. 



I append a Table giving the distances from the axis (h) at which 

 the light crosses each surface, numbered in order, and also the 



