254 Proceedings of the Royal Irish Academy. 



employed, if homogeneous ; always supposing that the aperture of the 

 objective used equals 180°. 



It is, furthermore, easy to show that if the incident light is as 



oblique as possible, the Kmit of visibility becomes = -, or about 0*0002 



millimeter, approximately = one one-hundred and twenty-five thou- 

 sandths of an inch. Photography enables us to advance a little further 

 into the unknown along this path, but the advance thus made is com- 

 paratively small. 



Finally, I have to apologise for presenting to the Academy so rough 

 a paraphi^ase of the work done in reference to this subject by Professor 

 Abbe, and by MM. JX^ageli, and Schwendener : my excuse is, that per- 

 haps that work may thus be made known to some earlier than it 

 otherwise would be. 



jS'otes added in the peess. 



I. — The diagrams (1) and (2) in the text are copied from Figures 

 125 and 128, at pages 222 and 226 of the 2nd German edition of 

 Schwendener and K'ageli's work on the Theory and Use of the Micro- 

 scope. Published by Engelman, Leipzig, 1877. 



II. — It has been pointed out to me that the incapacity of the 

 direct pencil for representing minute detail should be explained in 

 few words, as the explanation has been omitted by the writers men- 

 tioned above. An object can give rise to a visible shadow only when 

 the portions of the luminous wave which spread into the geometrical 

 shadow destroy each other wholly or in part by interference, in conse- 

 quence of the length of their paths differing by an uneven multiple 

 of half an undulation. As the light which passes by the edges of a 

 given microscopic object proceeds fi'om the luminous source, it is in 

 the same phase of undulation throughout at that passage, and conse- 

 quently the secondary waves which bend into the geometrical shadow 

 cannot interfere to produce a real shadow, unless the difference of the 

 length of the lines imagined to be drawn from the edges of the object 

 to any point of its geometric image be at least half an undulation of 

 the light employed. If it is less than that quantity, the whole of 

 the geometric shadow will be filled with light, and the object will be 

 invisible. This applies to objects of any form or magnitude disposed 

 in any manner whatsoever, and to orcdnary, as well as to assisted 

 vision. 



