10 REPORT— 1841. 



solid, bounded by two parallel planes, each of which is, and has been, for an 

 indefinite time, kept at the same uniform temperature throughout, repre- 

 sented for the one plane by a and for the other by h, is expressed by the 



following equation : u = a + z ; where v is the temperature at the 



e 



distance z from that plane whose temperature is a, and e is the distance be- 

 tween the planes. (Fourier, Theorie de la Chaleur, Art. 65.) 



2. A very small square prismatic bar is kept heated at one end until the 

 different parts of the bar have acquired a permanent temperature. That 

 temperature, or rather, as it actually is, the excess of temperature above that 

 of the surrounding medium, is represented by the equation, 



wt= Ac V A7+Be VJ7; 



where x is the distance of the point whose temperature is v from the heated 

 extremity of the bar, / is a side of the section, and h and k the coefficients 

 of radiation and conductivity of the bar. 



Cor. If the bar be supposed very long, B must be equal to 0, and v = 



„ /2h 



Ae~'^V*T. _ 



Here A represents the heat of the extremity, and a / _ is a quantity 



which must simply be determined by experiment. (Art. 76.) 



3. The permanent state of temperature of a ring is represented hj v = 



A a~ + B a , where x is the distance of the point under consideration 

 from some fixed point, measured along the arc which passes through the 

 centre of the generating circle. (Art. 106.) 



Cor. If points be taken at equal distances along the axis of the ring, the 

 ratio of the sum of the temperatures of the first and third to the temperature 

 of the second, is the same, whichever point be taken first. (Art. 107.) 



4. The temperature at any point of a ring which has been heated at one 

 point to a stationary temperature, and is then suflPered to cool, is represented 



I. r. "''"ivTri cosa;e "''' cos2a;e"" ^^'^^ cos3a;e~^^** 



byt; = 2e M|- -^^-^ + "S^+T " ^mHT 



\ (Art. 242.) 



5. As the time increases, the law of temperatures in a ring tends to become 

 such that the sum of the temperatures at the opposite extremities of a dia- 

 meter is equal to 2 « ~ ' ; which shows that the sum is the same at the ex- 

 tremities of whatever diameter we estimate them. (Art. 245.) 



II. Libri's hypothesis. That the interior conduction follows Newton's, and 

 the extra- radiation Dulong and Petit's law. The author has only applied 

 his analysis to the motion of heat in a ring. The conclusion to which he 

 arrives is the following : 



6. If a ring be heated at one point and then left to cool, the sum of the 

 temperatures at a given instant at the two extremities of a diameter is the 

 same for every diameter that may be taken. 



This result, which is only approximate, is not adapted for testing the truth 

 of the theory. It is, however, quite independent of any considerations re- 

 specting the mode by which the equations may be integrated. 



The author of the present report has applied Libri's hypothesis and method 

 to the solution of the same problem. He finds that 



7. The effect of Dulong's law is, that the velocity of cooling diminishes 



