178 PROCEEDINGS OP THE AMERICAN ACADEMY 



preciable. But even in this case, only ten increments are required in 

 order to produce an error of measurable magnitude. 



A simple and direct way to determine the degree of precision with 

 which measures under the microscope may be made, is to compare 

 measurements of the same space made by different observers and 

 under different conditions. I may get results which show an agree- 

 ment inter se quite within the limits of the accuracy required, but 

 which are yet wide of the truth. But if another equally skilful 

 observer obtains substantially the same results from a series of meas- 

 urements made under entirely different conditions, the inference of 

 their general correctness may be drawn with tolerable safety. 



In carrying forward this investigation I was fortunate in securing 

 the co-operation of Professor Edward W. Morley, of Hudson, Ohio, 

 whose paper will be found on page 16-4 of this volume of the Proceed- 

 ings. 



The rulings selected for joint measurement, are described as fol- 

 lows : — 



Plate I. consists of eight bands. The first three bands are composed 

 of twenty-six lines each. The distance between the lines is ^|<y of 

 an inch. The remaining five bands are composed of twenty-one lines 

 each, the distance between the lines being ^-J^ of an inch. All the 

 rulings of this plate involve the periodic errors which belong to the rul- 

 ing screw. 



Plate II. consists of three bands of very heavy lines, each band 

 being composed of twenty-six lines. The interval between the lines 

 is the same as in the corresponding three bands of Plate I. The 

 lines are filled with graphite and are mounted in balsam. In this plate 

 the errors ivhich are a function of one revolution of the screw were 

 corrected during the process of ruling. 



Plate III. consists of 101 lines, separated by an interval of -%£$-$ 

 of an inch, and freed as nearly as possible from errors of all kinds. 



Plate IV. consists of 21 lines, separated by an interval of Ts\j mm ', 

 corrected for systematic errors. 



The results given in the following tables under the head " Corr." 

 represent the corrections which must be applied to each space of a 

 given band in order to make it equal to a mean of all the spaces. 

 They are expressed in millionths of an inch, except in Plate IV., in 

 which the unit is one hundred-thousandth of a millimeter. The results 

 given under the head 2 represent the accumulated errors reckoned 

 from the first line of each band. In Plate I. the values given were 

 formed by successive additions of the individual errors. In Plates II., 



