414 Transactions of the Society. 



the unaided eye ; this tangent, when multiplied by a million to get rid 

 of the decimal point, will be distinguished by the letter v. Secondly 

 find out the tangent of the similar angle with a Microscope, care being 

 taken that the minimum visible is not the minimum resolvable with 

 the given objective ; in other words, the power must be so adjusted that 

 the minimum visible is within the grip of the apertui'e of the objec- 

 tive ; this tangent, when multiplied by a million, will be known by m. 

 Thirdly, the " lag " is the difference between the quantities represented 

 by m and v. 



Of course it is necessary that the observer measuring the various 

 objectives should determine the value of v for his own sight. Let us 

 suppose that an observer can divide the one-hundredth of an inch at 

 10 in. ; then by a Microscope with a power of 100 he ought, if there is 

 no "lag," to be able to divide the one ten-thousandth of an inch at a 

 10 in. projection distance. In this case v is 1000, and m also 1000, 

 the " lag " or the difference between these values being 0. If, on the 

 other hand, the Microscope is capable of resolving only 8000 lines to 

 the inch with the same power, m will be 1250; and as v will be 

 1000, as before, the ■' lag " will be 250. In 1875, while experimenting 

 with some telescopes, 1 found that if a certain grating could just be 

 resolved by the unaided eye at a distance of say 20 yards, the same 

 grating could always be seen with a telescope of a power, say, 10 at a 

 distance greater than 200 yards. Similar experiments were performed 

 with many telescopes of various powers, the powers and distances 

 being all very accurately measured, and in every case the telescope 

 resolved the grating at a distance greater than the naked eye distance 

 multiplied by the telescope's power. This shows that, so far as the 

 telescope is concerned, the " lag" becomes a negative quantity ; if then 

 we meet with a similar experience with respect to a Microscope, we 

 need not be surprised. 



We have seen above that the " lag " in microscopic vision may be 

 expressed by the formula tan M — tan V ; it will now only be neces- 

 sary to explain the practical method of using this formula for non- 

 mathematical readers, and to give a table of values obtained from 

 objectives tested in this manner. To find tan V, set up any suitable 

 grating, for example an ivory scale divided into fiftieths of an inch 

 would answer the purpose, and measure the greatest distance at which 

 the lines can just be separated ; a bright day must be chosen for per- 

 forming this experiment. Suppose that one-fiftieth of an inch can just be 

 divided at 45 in., then tan V is 1/50 or 0-02 ; divided by 45, this 

 equals - 000444. Multiplying this by one million, we obtain the 

 value of v as 444. To find tan M, divide the magnifying power by 

 ten times the number of lines to the inch on the test-plate resolved by 

 the Microscope. Example : — with a power of 280 the number of lines 

 to the inch resolved is 45,000, then tan M will be 280 divided by 

 450,000. This is equal to ■ 000622, which' multiplied by a million 

 = m = 622. In this case the " lag " m - v = 622 - 444 = 4-178. 



