364 Mr. Louis Bell on the Absolute 



moved along about the length of the run, and the process 

 repeated till the whole grating had been gone over. The 

 variations in the micrometer-readings then gave the variations 

 in the length of n spaces in different parts of the grating. 

 The only assumption involved was that the variation in the 

 different sections did not amount to an entire space, an 

 hypothesis quite secure in gratings with spaces as large as 

 those employed. It was thus possible to determine quite 

 accurately the variations in the grating-space throughout the 

 whole grating. 



It should be noted that, since these variations may be of 

 almost any kind and magnitude, the errors produced by them 

 will not in general be eliminated by combining the results 

 obtained from several gratings. It may happen that the 

 gratings used by one experimenter will have errors that will 

 counterbalance each other, while those used by another will all 

 have errors of the same sign. For instance, by the merest 

 accident the gratings used by the writer gave nearly identical 

 results corrected and uncorrected, while those used by Peirce 

 uniformly required a reduction in the resulting wave-length. 

 The number of gratings used by a given investigator is, how- 

 ever, so small that the errors will very seldom be eliminated; 

 while no combination of the results obtained from different 

 orders of the same grating can produce any useful effect 

 whatever. 



Each of the gratings used in this research was examined 

 minutely by the above methods, and in each was foiud an 

 abnormal portion of one sort or another. Of eight gratings 

 which I have calibrated all have shown a similar error; and of 

 more than twenty which I have examined in the spectrometer 

 only one (grating III.) failed to show an abnormal section at 

 one end. Since this is the commonest form of the error in 

 question, it is but natural to inquire why it cannot be avoided 

 by covering the defective end. The reason is simple enough. 

 By stopping out the defective portion the grating is reduced 

 to an incommensurable length, which enormously increases 

 the difficulty of measuring it. A grating which is in length 

 some convenient submultiple of a metre is easy to measure 

 with a comparatively high degree of exactness; but one which 

 is, say, twenty-seven millimetres long, is exceedingly difficult 

 to measure accurately, since it involves a long micrometer run 

 or the errors of subdivision down to single millimetres. It is, 

 therefore, better to use the full aperture of the grating and 

 find A by calibration. 



In calibrating the gratings used, I divided I. and II., which 

 were thirty millimetres long, into six sections of 5 million, 



