480 Dr. C. Y. Burton on a Physical Basis 



vation which has no direct connexion with the above, but 

 which may probably be of immense importance in the theory 

 of fluorescence, namely that fluorescein and its ammonium 

 salt, although they have quite different absorption spectra, 

 yet give out qualitatively exactly the same kind of fluorescent 

 light. We have then here, so to speak, two dissonant strings 

 of different materials ; and there is placed before us with 

 considerable emphasis this remarkable peculiarity of fluore- 

 scence as opposed to acoustic resonance, that the wave-length 

 of the light exciting the sympathetic vibrations is, within 

 certain fixed limits, of such slight importance. 

 Hamburg, Oct. 1889. 



LYII. On a Physical Basis for the Theory of Errors. 

 By Charles Y. Burton, D.Sc* 



1. ~TN deducing a law of error, two courses seem open to 

 J- us. We may make our assumptions as general as 

 possible, so that our results shall have the widest application, 

 and shall in the long run approach most nearly to the truth ; 

 or we may treat each separate case as a special problem in 

 probability, taking account of all that we know concerning 

 the actual conditions. 



I shall here endeavour to illustrate the latter method by 

 means of some examples ; proceeding next to the resultant 

 law of error when two or more elements are combined which 

 are independently subject to error. The most advantageous 

 combination of fallible measures will then be shortly discussed, 

 and, finally, subjective or personal errors will be considered. 



2. Suppose that we are given a series of numbers, known 

 correctly to any required number of places, and that from this 

 we write down the same series correct to four places. There 

 will be no uncertainty in the operation unless the digit in the 

 5th place is 5, and all the remaining digits zero ; and (in 

 general) the chance of this occurring is indefinitely small. 

 The limits of possible error are obviously + *00005, and all 

 errors between these limits are equally probable, unless from 

 our knowledge of the series we have a priori evidence to the 

 contrary. The curve of error (as one may call it) is thus a 

 finite straight line AB (fig. 1), parallel to the axis of errors 

 L M, and bisected by the ordinate of no error, N. If the 

 original table is carried only to (say) 5 places the case will 

 be somewhat changed. About ^ of the series of numbers 

 will have 5 in the 5th place of decimals ; the remaining -fa 

 * Communicated by the Physical Society : read November 1, 1889. 



