142 



THE AMEKICAN MONTHLY. 



[August, 



If c o can be determined, and from 

 that the distance from c to the front 

 of the objective (the working dis- 

 tance) be taken, o will be determined 

 with reference to the front of the ob- 

 jective and its position definitely fixed. 

 Now to find c o we recur to the equa- 

 tion above. 



a b' c o 

 a b o c 

 and by adding one to each side we 

 have — 



a b' CO, 



^^~X + ^ = — ^ ^ 5 

 a o CO 



a b' -\-ab c o-\- c o c c 



or, 



a b 



or, CO = 



CO CO 



c c'Xa b 

 a' b' -{- a b' 



Now, cc\ a b, and a b' are quanti- 

 ties which may practically be meas- 

 ured, and thus a definite value of c 6> 

 is obtained, and from that ois fixed. 



Now, theory l,eads us to expect 

 what experiment bears out, viz., that 

 as the position of a b varies, and with 

 it that of a' b\ the point o does not 

 remain quite stationary. In some 

 objectives it moves slowly back as the 

 object is approached to the objective, 

 and in others slightly nearer the front. 

 But since we wish to have o deter- 

 mined when c o = lo inches, that 

 calls for one definite, fixed position 

 of a ^, and for a fixed value of the 

 distance c o. But c o cannot be fixed 

 unless both c and o be known ; that 

 is, unless o be already known. It 

 would, then, appear that the problem 

 is impossible since it calls for the re- 

 sult as a necessary condition to the 

 solution. This is true, strictly speak- 

 ing, but, practically, as close an ap- 

 proximation as may be desired is 

 readily attainable. 



The change in o for a change of an 

 inch or two in a b' will be probably 

 less than the one hundredth of an inch, 

 so that if the lengths be so adjusted 

 that c c' equals about lo inches plus 

 the focal length of the objective, the 

 distance c o will come out differing 

 from lo inches by less than an inch, 

 and, probably, by less than a tenth of 



an inch, and o will be as accurately 

 determined as ordinary measurements 

 are made. If, however, greater ac- 

 curacy is desired, the lengths maybe 

 readjusted so that from the determined 

 position of o to c shall be lo inches. 

 Then, making the measurements 

 again, a new value is found for c <?, 

 which will differ slightly from the 

 first one, and be a still closer approxi- 

 mation. The position of o found from 

 this value of c o will differ from its 

 true position when c o h lo inches, 

 by less than one thousandth of an inch, 

 probably, which is far within the 



I limit of error in such measurements. 



j So much for the theory. In the 

 practical application, a stage mi- 

 crometer is placed in position, and one 

 of its divisions serves for a b. The 

 eye-piece is removed, the tube placed 

 vertical, and a screen of ground glass 

 or thin oiled paper is placed over the 

 end of the tube on which to receive 

 the image «' b' . 



Let us suppose that the objective to 

 be measui-ed is one rated as a J-inch ; 

 then bring roughly to a focus, and 

 next adjust the draw-tube until the 

 distance from micrometer to screen is 

 lof inches within, say, an eighth or a 

 tenth of an inch. Then focus care- 

 fully until the lines are distinctly de- 

 fined on the screen. Then measure 

 the distance from the micrometer to 

 the screen or c c . This is found, we 

 will suppose, to be 10.84 inches. 

 Next, measui-e the value of a division 

 on the screen. To do this as accu- 

 rately as possible, measure the length 

 of as many divisions as can be seen, 

 and divide by the number. This will 

 divide, by the same number, the error 

 in making the single measurement. 

 This gives a b\ which in the case we 

 have supposed equals .1125 inch. 

 The value of a b, that is, of a divi- 

 sion on the micrometer, is .01 inch. 

 Hence we have, making the proper 

 substitutions, 



^^ _ 10.84^.01 __ 

 .1225 

 Taking this from 10.84 ^^ ^^^ c o = 

 9.925-)-. Next, measuring the work- 



