12 REPORT — 1841. 



temperature, or rather the excess of temperature above that of the sur- 



d V 

 rounding vacuum ; then since Fourier's equations hold good, we have — = 



— av, where « is a quantity which depends on the radiating power. 

 But by Dulong and Petit's formula, 



dv a V 



"d6-i3\e_^' 



and log V = —, log A (1 — a~ ) 



^ 10 log^ a *= ' ^ 



jS logg a 

 V « =A(l-a~% 

 If /3 log a = a, this equation is reduced to 



To the collection of formulae which we have extracted from the writings of 

 Fourier, Poisson, and Libri, we may add the following, which apply to ex- 

 periments of the simplest kind. 



9. On Libri's hypothesis, the permanent state of temperature in a small 

 uniform bar of indefinite length heated at one end is represented (approxi- 

 mately) by the equation 



A- log^ a _9„r 



v—Ke ■^ + g 



The demonstration of this proposition is as follows : 

 The equation of motion is 



- — = X (cp — 1), where a = \/ 1*165 according to Dulong and Petit. 

 dx- ^ 



By expansion, — = \^v log^ « + ^ i^^Se «)" f 



o2 

 = 9""" + "^ loge « • '"- nearly. 



Assume v = A e~^ + V, 



CI CC" it 



d~ V o^ 



consequently ^— J- =5^2 V + -^ log^ a . A- e"^^^;^ 



whence V = :^L2^?ile- 2-7^ 

 6 



AMog a 



and V = Ae-S^ + -5£_ e-2^cr_ 



b 

 10. On Fourier's hypothesis, the temperature at the extremity of a like bar, 

 when its length is not sufficiently great to be regarded as infinite, is t;' = C u, 

 where u is the temperature of the heated extremity, and C is a constant, not 

 depending on u. 



The equations of motion are 



d'~v 2 h J dv , h * , . „,, 



-j—l = -T-T V ; and -z — \- —v = 0^) at the extremity of the bar. 



■* Fourier, arts. 74 and 124 ; Kelland, arts. 53 and 57. 



