Comparators for Measures of Leiigth. 215 



errors of the ruled plate from line i to any point up to line 

 100, but such measures involve the errors of the comparing^ 

 screw. In the second case, the accumulated errors are 

 measured in the same way from line 100 to line i. But if we 

 subtract the measures from line 1 to line 100 from the corre- 

 sponding measures from line 100 to line i, the differerue will 

 give twice the accumulated errors at any point, indepemient of 

 t/u errors of the comparing screw. The only exception to this 

 rule is found when the cur\'e of errors takes a wave form. In 

 a general way, this will be the case when the maximum error 

 falls near line 25, and the minimum near line 75. 



As an illustration of the character of the work which may 

 be done with a comparator of this form, I give the measures of 

 five standard micrometers, ruled at different times. As these 

 micrometers are somewhat different in form from any with which 

 the writer is acquainted, a brief description will be necessarj-. 



I. A half inch is divided into 50 equal parts, the ist, 25th,, 

 and 50th spaces being again subdivided into 10 equal parts. 

 The length of the lines is about one-eighth of an inch, the 5th 

 and loth lines being a little longer. 



II. After arranging the position of the ruling carriage, so 

 that the lines of the second series of graduations should begin 

 near the point where those of the first end, coincidence is made 

 mechanically with the first line of the series already ruled. 

 For a short distance the ruling point goes over the same 

 ground twice. A centimeter is then subdivived into 10 equal 

 parts. The ist, 5th, and loth spaces are again subdivided into 

 10 equal parts, and one of the middle subdivisions is still 

 further subdivided, giving .01 mm. Xear by is a band of 21 

 lines, each space being equal to .001 mm. 



The first five columns of the following table give the 

 number of divisions of the comparing screw corresponding to 

 each space of .01 inch. The values for Xo. 7 are the mean of 

 two readings. The remaining values represent single measures. 

 The mean values given in the sixth column, without doubt, 

 nearly represent the actual errors of ruling, the accidental 

 errors of measuring being nearly eliminated in taking the 

 mean of the corresponding measures of the separate plates. 



The last column represents the accumulated errors at ever}* 

 point between line i and line 50, expressed in millionths of an 

 inch. With the sign given, the values represent corrections to 

 the corresponding spaces: 



