OP ARTS AND SCIENCES. 165 



Two bands on the first plate, and one each on the second and third 

 plates, were measured twice. The probable difference of two measure- 

 ments of the same interval was found to be one three hundred and 

 fifty -nine thousandth of an inch ; from which the probable error of 

 a single measurement may be presumed to be about two millionths of 

 an inch. It happened that thirty-five of the differences between two 

 measurements of the same space were less than the probable difference 

 as computed by the usual formula, and thirty-five were greater. 



The measurements on the third plate were more difficult than the 

 other, partly because the lines were too fine for the easiest work, and 

 partly on account of fatigue. They are, therefore, less satisfactory 

 than the measurements on the other plates. The outer lines of some 

 bands on the second plate were also troublesome, and the results for 

 two or three not so good as for other spaces. 



Mr. Rogers made measurements of the same plates, which he 

 prepared for publication without knowing my results, but after the 

 original micrometer readings of my measurements had passed beyond 

 my control. Of his measurements I know nothing at the time of 

 writing the following tabular results. By concert with him, my re- 

 sults are tabulated in the form adopted by him, for ease of compari- 

 son. My numbers for the spaces measured increase in the direction 

 of the arrows on the ruled plates, if I have made no mistake, and also 

 my numbers of the bands. The numbers of the plates are those 

 written on them by Mr. Rogers. In the third plate I measured only 

 spaces composed of five of the spaces of one twenty-four hundredth of 

 an inch as ruled ; the difficulty of the measurement of so faint lines, 

 as well as the fear of incurring a return of a certain slight difficulty 

 with one of my eyes, from which recovery was not then complete, led 

 me thus to abridge this part of the work. It is to be regretted that 

 this plate was not taken in hand earlier. 



The figures in the columns of individual and accumulated errors 

 represent millionths of an inch composed of twenty-four revolutions 

 of the screw of Mr. Rogers's ruling engine. But in the case of the 

 fourth plate they represent hundred-thousandths of a millimetre of a 

 similar standard. 



