352 MR. A. E. TTJTTON ON A COMPENSATED INTERFERENCE DILATOMETER. 



length of the solid substance at is represented by L 0) and the length at f by ~Lt, 

 the nature of the change of length is adequately represented by the formula 



Lt = L (l + at + It 2 ). 



If be the absolute coefficient of linear expansion, then the constant increment 

 per degree is A/A, and their relations to the constants a and b of the above 

 formula can be ascertained at once by successive differentiation with respect to the 



temperature. 



= a + 2bt, Aa/A = 26. 



The mean coefficient of expansion between and t is therefore a -\- bt; but the 

 true coefficient at any particular temperature t, and also the mean coefficient between 

 any two temperatures whose mean is t, is a + 2bt. 



Hence in order to be able to determine the true and mean coefficients, and the 

 increment per degree, which together afford full information as to the nature of the 

 expansion, it is only necessary to ascertain the constants a and b in the general 

 expression above quoted for L. 



In this formula ~Lt and t are known, and there are three unknown quantities, 

 L , a, and b. To determine them three equations, for three different temperatures, 

 are required. The data derived from the observations at the ordinary temperature, ^ 

 (about 10), at the first higher limit, t z (about 70), and at the highest limit, t 

 (about 120), enable the required three equations to be compiled. For L^ is the 

 length measured at the ordinary temperature by means of the thickness measurer ; 

 L> is L[ -\-f-i \ X ; and L^ 3 is L^ -f-,/7 \ X.. We have then the three equations 



3 



Lt, = L (l + at 2 + bfi), 

 Lt 3 = L (l +a 3 + bfy. 



By subtracting respectively the first from the second, and first from the third 

 equations, we obtain a pair of equations which, by complementary multiplication 

 with the coefficients of a and b respectively, enable each of these constants to be 

 eliminated in turn and the other to be obtained in terms only of L and the known 

 quantities. On substituting these values of a and b in any of the three fundamental 

 equations L is at once obtained, and its substitution in the expressions for a and b, 

 just referred to, affords the desired numerical values of these constants. 



The three expressions for a, b, and LO, in the form actually employed by the 

 author in the reductions, are as follows : 



