ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 109 



In any system of infinitely thin lenses in contact, the number of 

 lenses required to bring the greatest possible number of light-rays of 

 different degrees of refrangibility to a common focus is the same as 

 the number of different powers of A contained in the dispersion 

 formula employed. 



The method made use of in arriving at this result has been 

 adopted, because it brings out clearly the geometrical relations of the 

 problem. The result itself is evident from a mere inspection of 

 equation (5), which cannot possess more real positive roots than it has 

 independent auxiliaries, D, E, F, etc. 



Colour Corrections of Achromatic Objectives.* — The following 

 abstract is published of a paper by W. Harkness : — 



1. From any three pieces of glass suitable for making a corrected 

 objective, but not fulfilling the conditions necessary for the complete 

 destruction of the secondary spectrum, it will always be possible to 

 select two pieces from which a double objective can be made that 

 will be superior to any triple objective made from all three of the 

 pieces. 



2. The colour correction of any objective is completely defined 

 by stating the wave-length of the light for which it gives the minimum 

 focal distance. 



3. An objective is properly corrected for any given purpose 

 when its minimum focal distance corresponds to rays of the wave- 

 length which is most efBcient for that purpose. For example : in an 

 objective corrected for visual purposes, the rays which seem brightest 

 to the human eye should have the minimum focal-distance ; while in 

 an objective intended for photographic work the rays which produce 

 the greatest effect upon silver bromo-iodide should have the minimum 

 focal- distance. 



4. In the case of a double achromatic, the secondary spectrum 

 (or in other words, the diameter, at its intersection with the focal 

 plane, of the cone of rays having the maximum focal length) is abso- 

 lutely independent both of the focal length of the combination, and 

 of the curves of its lenses ; and depends solely upon the aperture 

 of the combination, and the physical properties of the materials 

 composing it. 



5. When the focal curve of an objective is known, and the 

 relative intensity, for the purpose for which the objective is corrected, 

 of light of every wave-length is also known ; then the exact position 

 which the focal plane should occupy can be readily calculated. 



Incidentally, it may be remarked that in an objective corrected 

 for photographic purposes the interval between the maximum and 

 minimum focal distance is less than in one corrected for visual pur- 

 poses. Hence a photographic objective has less secondary spectrum, 

 and is better adapted for spectroscopic work, than a visual objective. 



Verification of Objectives. — The editor of the 'Northern 

 Microscopist ' undertakes, for a nominal fee of Is. 6d., to verify 



* Bull. Phil. Soc. "Washington, ill. (1878-80) pp. 39-40. Smithsonian Misc. 

 Coll., XX. (1881). 



