of Air- and Mercmnj-Thermometers. 225 



The denominators of these coefficients being in arithmetical pro- 

 gression, indicates that the curve of exjiansion approximates closely 

 to a hyperbola, with the axes of coordinates parallel to the asym- 

 ptotes. We find, accordingly, hy drawing that curve through the 

 coordinate points at 0°, 100°, and 200°, that it will be found also to 

 pass within j-^th of a degree of the point at 300°. 



The data as to the expansion of mercury in glass are as follows : — 



Between 0° and 100° the rate of expansion is _ .„„ ♦ 



o4o0 



Between 0° and 200° „ „ -^. 



6378 



Between 0° and 300° „ „ -1-. 



6318 



From these we obtain the following volumes of glass : — 



0° 1-000000 



100° i-002586 



200° 1-005508 



300° 1-009120 



By considering the volume to increase, partly as the temperature 

 and partly as the cube of the temperature, we may represent the 

 curve of expansion exactly by an equation of the form 



By combining this with the equation for the hyperbola in which 

 mercury expands, the above formula for t,n in terms of ta was con- 

 structed. 



The original equation for the hyperbola is 



a—t^v—b^ =c2, 



in wliich the constants a, b, and &^ are determined from three obser- 

 vations. When t = 0, we have v=\, and the equation becomes 

 a\—b^ = c-. Let m =: absolute increase of volume from 0°; then 

 v—m=l, v — b=l +m — b, and c-=a—t{]. + m—b), and 

 c2— 1— 6'(a — l^^t 

 a — t a—t 



In the formula, B represents \ — b, and A represents a. 



The numerical values of B and A are fixed, because tm coincides 

 with ^„at 100", 



The expansibility of the glass corresponds with the maximum 

 observed by M. Pierre {Ann. de Chhn. vol. xv. p. 335). The mini- 

 mum is about three-fourths the maximum. If we adjust the formula 

 to the minimum expansion of glass, and compute the value of the 

 correction at 50°, we shall find that it does not sensibly differ from the 

 correction with thermometer glass of maximum expansion, the 

 amount of diiference being less than y'yjth of a degree. 



The alteration of the formula to conform to a change in the qua- 

 lity of the glass requires a change in the first term, which contains 



Phil. May. S. 4. Vol. 15. No. 'J'J. March 1858. Q 



