TABLE OF CONTENTS. XV 



CHAPTER IV. 



Of the linear and varied Movement of Heat in a ring. 



SECTION I. 



GENERAL SOLUTION OF THE PROBLEM. 



ART. PAGE 



238241. The variable movement which we are considering is composed of 

 simple movements. In each of these movements, the temperatures pre 

 serve their primitive ratios, and decrease with the time, as the ordinates v 

 of a line whose equation is v=A. e~ mt . Formation of the general ex 

 pression ... 213 



242 244. Application to some remarkable examples. Different consequences 



of the solution 218 



245, 246. The system of temperatures converges rapidly towards a regular 

 and final state, expressed by the first part of the integral. The sum of 

 the temperatures of two points diametrically opposed is then the same, 

 whatever be the position of the diameter. It is equal to the mean tem 

 perature. In each simple movement, the circumference is divided by 

 equidistant nodes. All these partial movements successively disappear, 

 except the first ; and in general the heat distributed throughout the solid 

 assumes a regular disposition, independent of the initial state . . 221 



SECTION II. 

 OP THE COMMUNICATION OF HEAT BETWEEN SEPARATE MASSES. 



247 250. Of the communication of heat between two masses. Expression 

 of the variable temperatures. Remark on the value of the coefficient 



which measures the conducibility 225 



251 255. Of the communication of heat between n separate masses, ar 

 ranged in a straight line. Expression of the variable temperature of each 

 mass; it is given as a function of the time elapsed, of the coefficient 

 which measures the couducibility, and of all the initial temperatures 



regarded as arbitrary 228 



256, 257. Remarkable consequences of this solution 236 



258. Application to the case in which the number of masses is infinite . . 237 

 259 266. Of the communication of heat between n separate masses arranged 

 circularly. Differential equations suitable to the problem ; integration of 

 these equations. The variable temperature of each of the masses is ex 

 pressed as a function of the coefficient which measures the couducibility, 

 of the time which has elapsed since the instant when the communication 

 began, and of all the initial temperatures, which are arbitrary ; but in 

 order to determine these functions completely, it is necessary to effect 



the elimination of the coefficients 238 



267271. Elimination of the coefficients in the equations which contain 



these unknown quantities and the given initial temperatures . . . 247 



