18 Transactions of the Society. 



ment. We may, therefore, take the following magnitudes to be 

 known. 



The optical length of the Microscope ; which may be written L. 



The equivalent focal length of the ocular; — written /^ 



The diameter of the speculum globule; — written g 1 . 



The equivalent focal length of the objective ; — written f 2 . 



The diameter of the object globule ; — written g 2 . 



The diameter of the dark patch upon the object globule ; 

 — written D. 



It will be evident on reference to the diagram, fig. 6, that the 

 apparent size of the second order image in the object globule of its 

 own darkened surface, which may be written d, is, 



— if; f 2 f 2 * * * ^ ' 



It is evident from this equation that the dimensions of the 

 test object (d) can be varied in two ways ; that is to say, we may 

 alter the size of the object globule or we may alter the size of the 

 speculum globule, and thus, by varying these two elements in the 

 combination, we can produce a black dot of any required dimensions 

 however small. Moreover, the mathematical law is one that works 

 out to a very convenient system in practice. If we alter the size 

 of the speculum globule the value of (d) alters according to a 

 simple proportion, so that we may write the above expression (2) 



d = Gg lm 



C being a constant ; if everything except the speculum globule is 

 left unchanged. We have thus the means of very gradually 

 altering the dimensions of the test object by substituting speculum 

 globules of slightly varying dimensions. 



If, on the other hand, we vary the object globule we, of course, 

 alter the value of D at the same time. In fact, D is itself directly 

 proportional to g 2 , therefore we may write the product 



g 2 B = Ci^ 2 . 



If we assume everything to remain unchanged except the object 

 globule, we may write equation (2) as follows : — 



a = ^>ig 2 . 



It thus appears that by changing the object globule we very rapidly 

 alter the size of the test image, and if we alter the size of both the 

 globules simultaneously, we get finally a value in the form 



d = 3 g 1 g 2 i . 



Under these conditions the size of the test image varies very 

 rapidly indeed. And thus with a comparatively small range of 



