96 America?! Quarterly Microscopical Journal. 



did not demand the reading of a quantity so minute as the 

 fourth part of a division. The remaining eight}^ measurements 

 were read to this quantity. The following table gives, in the 

 first column, the values of the differences between each pair 

 of measurements in ten-millionths of an inch. The second 

 column gives the number of times each difference occurred 

 among the eighty specified measurements, and the third col- 

 umn the number among the twenty. Computation will show 

 that the probable difference of a pair of measurements among 

 the eighty, is twenty-nine ten-millionths of an inch ; and for 

 the whole hundred measurements, thirty ten-millionths : 



Whether the smaller probable difference, obtained with 

 the object-glass of shorter focus, shows that the greater 

 amplification is better suited to accurate measurement, is 

 a point on which the writer has not formed a decided opin- 

 ion. Two or three years ago he obtained smaller probable 

 errors with the inch object-glass than with the quarter ; but 

 the experiments made to determine which should then be used 

 for a. given purpose were few in number. He has inclined to 

 the belief that the greater accuracy of the later series was, 

 more probably, not due to instrumental causes. 



From the probable difference of a pair of measures the 

 probable error of a single measure may be deduced by multi- 

 plying the former quantity by the square root of one half. The 

 propriety of this may be seen by observing that, since the 

 spaces measured in each series were professedly equal, the dif- 

 ferences in each column may be considered to be the differ- 

 ences between two measurements s'elected from a number of 

 measurements of the same space in pairs, one as much greater 

 as the other is less than the mean of this number of measure- 

 ments. Half of each difference would be the difference of a 



