14 RRPORT — 1841. 



2. From the second equation 



9+f 3 (5- +/)' ' 



4 2 



Also u the temperature at the heated extremity is of the form 2p A + — : 

 .'.A=-pq+ Vp-.q'i + qu 



A^ 

 and v' = 2 jtJ^ A -\ , in which p^ and q^ differ from p and q in not con- 



taining e^ , e~^ ; or may be deduced from them by writing b = ; 



••• V = 2p^{ ^p^q«-+qu -pq}+ {^tSl+JU^^llll, 



13. The expression for the permanent temperature at one extremity of a 

 bar heated at the other extremity, determined on M. Poisson's hypothesis, 



differs from the foregoing only in having - — A^ instead of °^ A- 



^ 6 



as the coefficient of e^ ^ ^* ' ^\ y-2m ^^ .^^^^^^ ^^ log^g ^^ ^^ ^^^^ ^^ 



3 6 



g—g \ —^) ^^(j 2 y instead of log^ a as the coefficient of the constant term in v. 



14. The expression for the same permanent temperature derived from the 



fourth hypothesis is 1 — a~ = C(l —«"""). 



15. When a bar, so large as that it may be considered infinite, is heated 

 at one end to the constant temperature a, and kept at the other to the con- 

 stant temperature b, as assumed in formula 1 , the state of temperature at any 

 point, as deduced from M. Poisson's theory, is given by the equation 



^_a+'-^(v"--a"-) + (a-b+'^ o^^^^) - = 0. 

 2 2 e 



The proof of this proposition is as follows : 



Since the bar is very large, the effect of radiation is zero (at least near the 



centre of the bar). Therefore the equation of motion is -7— • I -r^ )^^^' 



j,dv 



or « -7— = c. 



a z 



Now Poisson's approximates by supposing k' =.k -\- kmti. 

 Hence the equation is immediately integrable, and the result is as above 

 given. 



16. The corresponding equation due to the fourth hypothesis is 



-a -b 

 — V —a a — a 

 \—a =1— a +— z; 



the a which is affected with an index being Dulong and Petit's a. 



We have not thought it relevant to our present subject to point out the 

 difference which exists between hypotheses III. and IV.; as, however, we have 

 given only approximate formulae corresponding with the former hypothesis, 

 it may be proper to mention, that, for expressions not involving the time, the 

 essential difference between the formulae will be seen by the difference of 

 sign of the quantity which corresponds with m. It will be seen that on hy- 



