OF ARTS AND SCIENCES. 85 



A t -f B (' + &c. 



T= 100 



100 4 + (100)-- ! fi- + &c:. ' 



which is the same as for the weight thermometer. 



If the fixed points are 0° and t° instead of 0° and 100°, we can 

 write 



At + Bt*+Ct* + &c. 



*-=<{!+ «-»{!+ j+5'+i «}+*•} 



As T'and £ are nearly equal, and as we shall determine the con- 

 stants experimentally, we may write 



t= T— at(l' — t) (b — t)-\- &c., 



where t is the temperature on the air thermometer, and T that on the 

 mercurial thermometer, and a and b are constants to be determined for 

 each thermometer. 



The formula might be expanded still further, but I think there are 

 few cases which it will not represent as it is. Considering b as equal 

 to 0, a formula is obtained which has been used by others, and from 

 which some very wrong conclusions have been drawn. In some kinds 

 of glass there are three points which coincide with the air thermome- 

 ter, and it requires at least an equation of the third degree to repre- 

 sent this. 



The three points in which the two thermometers coincide are given 

 by the roots of the equation 



t (f — t)(b — t) = 0, 



and are, therefore, 



t = 



t = t' 



t = b. 



In the following discussion of the historical results, I shall take 0° 

 and 100° as the fixed points. Hence, i' = 100°. To obtain a and b, 

 two observations are needed at some points at a distance from 0° and 

 100°. That we may get some idea of the values of the constants in 

 the formula for different kinds of glass, I will discuss some of the 

 experimental results of Kegnault and others with this in view. 



