526 Transactions of the Society. 



Professor Abbe has given the following formula or theorem 

 capable of general application for the solution of the several 

 problems relating to intensity of illumination : — 



" Leaving out of calculation any loss occasioned by reflection 

 from the lenses of the microscope, the brightness of the completely or 

 partially transparent parts of the object is exactly the same as that 

 with which it would appear to the naked eye looking through a 

 diaphragm aperture immediately in front of it, the size of which 

 aperture corresponds with the area of the image of the light- 

 giving source, as seen at the eye-point of the microscope: the 

 object being supposed to be viewed in its magnified dimensions 

 against the light of the primary source as a background. 



"As direct corollaries from this theorem it results — 1. That the 

 brightness of the microscope picture can in no case exceed that 

 with which it would appear to the naked eye. But that for each 

 angular value of the image-forming pencil, whether limited by the 

 objective aperture or by the area of illuminating surface, there is a 

 given amount of amplification, heloiv which the brightness of the 

 image is always equal to that of the object as seen by the naked 

 eye. And this amount of amplification is reached when the image 

 at the eye-point has the same diameter as the pupil of the eye. 



" 2. The intensity of brightness increases with the increase of 

 angular aperture of the objective, so long as that aperture is not as 

 large as that of the illuminating cone. But when the objective 

 aperture is larger than that of the illuminating cone (and this is the 

 usual case), the brightness diminishes. This happens with all the 

 higher amplifications. 



" 3. Ceeteris iMvibus, the brightness of image is inversely 

 proportional to the square of the linear amplification when the 

 area of eye-point image is smaller than the pupil opening. Whether 

 the amplification be obtained by a strong objective and weak ocular 

 or by a weak objective with strong ocular, the amount of light with 

 each amplification is the same. And all objectives whose aperture 

 exceeds the maximum angular value of the incident light-cone 

 transmit an equal amount of light when an equal amplification is 

 obtained by using oculars which equalize the magnifying power. 

 The appearances which seem to contradict these corollaries are 

 explained by the circumstance that all differences of sharp and 

 clear definition are unconsciously interpreted as differences of 

 brightness." * 



Finally, attention must be directed to the physiological conditions 



* The above extract is quoted from Professor Abbe's article in vol. ix. of 

 Schulze's ' Archiv fiir Mikr. Anat., ' entitled, " Contribution to the Theory of 

 the Microscope," published in 1874. The, tiieory above enunciated with its corol- 

 laries is contiriiitd Ijy Helmholtz, and agrees exactly with the results stated in the 

 treatise of Hclmhnlfz on the capacity of performance of the Microscope translated 

 by the present wr.ter for tlie Bristol Naturalists' Society's ' Proceedings.' 



