420 Transactions of the Society. 



The remaining conclusion of Helmholtz' paper depends entirely 

 upon this assumption concerning the limit of resolving power, and 

 must stand or fall with it. Helmholtz points out, as has been 

 already shown, that even with a divergence angle of 90° the anti- 

 point is not infinitesimal but has a false disc, the semidiameter 



of which by equation (5a) (p = ^ — : — ^ j is equal to (^) half a 



wave-length of light. Now, if such an image were magnified the 

 false disc would be magnified in the same proportion, and in that 

 case the smallest possible antipoint in a magnified image would be 

 proportionately larger than this smallest possible antipoint in the 

 object-plane. But he rightly says it does not make matters any 

 better to replace this minute image by a minute object, for although 

 the object may be more finely resolved in fact than the image for 

 which it has been substituted we cannot look at; the object itself. 

 We can only look at its magnified image, and, as we have seen, its 

 magnified image must be formed by beams of light having a 

 divergence angle so reduced that they yield antipoints upon the 

 same scale as the magnified image of the objective antipoint first 

 supposed. Now, it makes no difference in the result whether the 

 antipoint is formed on the stage and magnified in the same pro- 

 portion as the image, or formed in the image itself on the same 

 scale of magnification. Therefore it comes to this in the end. 

 We can determine the ultimate limit of resolving power in terms 

 of the object by ascertaining the dimensions of the smallest detail 

 that can be discerned in the image, and measuring its conjugate 

 image projected on the stage. By the above computation this 

 conjugate image is directly calculated, and thus Helmholtz came 

 to the conclusion that the smallest object which can be resolved 

 even by a perfectly corrected and ideally perfect lens, must be not 

 less in diameter than half a wave-length of the light by which 

 it is seen. 



Passing now from Helmholtz' paper, I desire in conclusion to 

 draw your attention to one or two practical deductions from the 

 Helmholtz theory. 



Oscillating Screens. 



(1) First let me refer to the inconveniences which result from 

 the progressive reduction of the divergence angle as the magnifica- 

 tion of the image increases. Helmholtz has pointed out that this 

 causes all obstructions in the eye or in the upper part of the 

 instrument, to throw very black and conspicuous shadows, so that 

 even the smallest speck of dust upon the eye-lens for example 

 will be projected so as to become a prominent blemish in the image. 

 The reason of this is that when the wave-front is much reduced in 



