186 PROCEEDINGS OP THE AMERICAN ACADEMY 



The average deviation is 19 teii-millionths of an inch, and there are 

 only 7 cases in which the disagreement exceeds 2 million ths of an 

 inch. 



(g) The relative advantages of the eye-piece micrometer, the filar 

 micrometer, and the screw comparator, for narrow intervals, are nearly 

 equal, as will be' seen from the following comparison of the individual 

 values derived by each method of observation, with the normal values 

 found from the equation 



n = — 23.8 sin x -f- 5.4 cos x — 0.1 sin 2 a; -J- 1.0 cos x, 



which represents the mean curve for the first band of Plate I. 



Number of millionths, 012 3 456789 10 



Eye-piece micrometer, 78350101000 

 Filar-micrometer, 42352151200 



Merz screw, 42425222011 



Number of 

 cases, 



(h) It appears from this investigation that it is possible to reduce 

 the errors of a precision-screw for short intervals to about one hun- 

 dred-thousandth of an inch by applying the corrections derived from 

 the equation which represents the periodic errors. Since the rejection 

 of oil as a lubricant, the errors have been considerably reduced. 



(i) In a meridian circle having a diameter of 30 inches, one second 

 of arc is equal to .0000727 of an inch. It appears, therefore, from 

 this investigation, that, even if the attached microscopes have the same 

 power as those used in this investigation, the ultimate limit of accuracy 

 in the matter of bisection and reading only, must be at least 0."05. 

 But the microscopes of the meridian circle of Harvard College Obser- 

 vatory magnify only 51 diameters, while the magnifying powers used in 

 this series of measures were 194, 290, 560, and 870. Moreover, this 

 limit has reference only to repeated readings of the microscopes for 

 the same position of the instrument. It has, therefore, only a relative 

 value. When, in addition to the errors of simple pointing and reading, 

 we take into account the accidental and the systematic errors of di- 

 vision in the graduated circles and the outstanding errors always found 

 in measures of large arcs of a circle, the present limit of precision can- 

 not fall much below 0."2. 



Since the completion of this investigation a further opportunity of 

 comparing the results of measures of the same intervals by different 

 observers has occurred. Through the kindness of Professor George 

 F. Barker, of the University of Pennsylvania, I obtained the loan of a 



