316 



used (he foriiuila by which ihe length of the l)ar at zero is derived 

 IVoiii the nieusiirements. 



During tlie nieasui-ements at Breteiiii tlie temperatures of the bar were, 

 according to Benoit's statements,, not vei^ different from 15°. I have 

 not here at my disposal the data of the exact values of the tem- 

 peratures dui-i ng the measurements at Delft, but I know that they 

 pi-esented no great deviations and, if I am not mistaken, the extreme 

 differences from the mean, about IS'^. wei-e not greater than about 

 two degrees. We can therefore combine the observations at Breteuil, 

 and also those made at Delft each into a mean result, at a mean 

 temperature, and we then obtain the following equation, in which 

 tlie letters without a dash indicate the values determined at Breteuil, 

 those with a dash the values determined at Delft: 



L, -l:=- aL, r 4- a'L: T + 4(/„ - ^o') + m - m ^s-s. 



In these formulae L is the length of the bar, T its mean tem- 

 perature during the measurements, « the adopted coefficient of expansion 

 of the bar, I the length of the comparison metre, / its mean tem- 

 pei'ature, /i the mean expansion of the metre for 1°, aS the difference 

 of the length of the bar and the fourfold of the length of the com- 

 parison metre, determined by means of micrometrical measurements 

 with the microscopes; T and o, as indices of L and / indicate the 

 temperatures to which these lengths have been reduced. 



The differences of the temperatures T— T' =r ATand t—{ = A^, 

 and also the differences in the adopted coefficients of expansion 

 /? — ^ =z t\^ are small, and for the value of the coefficient of ex- 

 pansion « and a the same value has been adopted in Breteuil and 

 in Delft; the last of the three equations may therefore be put 

 approximately into the following form : 



X« — /.;'=AL = — «LA7^-f4A^ + 4^A/ + 4i;A/? + ,S— .S'. 



When A 7', A;', A/, A.i and S — S' have their exact values, LL 

 is zero; the value 8'^.7 for AL found from the observations is there- 

 fore only a function of the errors in those values, and putting on 

 the first side of the formula A/>:=:8".7 the quantities on the second 

 side represent those errors. We will consider each of the terms 

 separately. 



1. aLtxT. T and T' have been determined in the same manner 

 by readings of the thermometers laid on the surface of the measur- 

 ing bar within the thick aluminium case; the temperatures in both 

 comparatoj'S were fairly constant, and the value of a is small; aL 

 for 0°.l is about 0^.7. In view of the great value of t\L, we may 

 therefore neglect that term, 



