XVI TABLE OF CONTENTS. 



ART. PAGE 



272, 273. Formation of the general solution : analytical expression of the 



result 253 



274 276. Application and consequences of this solution .... 255 

 277, 278. Examination of the case in which the number n is supposed infinite. 

 We obtain the solution relative to a solid ring, set forth in Article 241, 

 and the theorem of Article 234. We thus ascertain the origin of the 

 analysis which we have employed to solve the equation relating to con 

 tinuous bodies 259 



279. Analytical expression of the two preceding results .... 262 

 280 282. It is proved that the problem of the movement of heat in a ring 



admits no other solution. The integral of the equation -^= k -=-? is 



dt dx* 



evidently the most general which can be formed . 263 



CHAPTER V. 



Of the Propagation of Heat in a solid sphere. 



SECTION I. 

 GENEBAL SOLUTION. 



283 289. The ratio of the variable temperatures of two points in the solid 

 is in the first place considered to approach continually a definite limit. 



This remark leads to the equation v=A g-J&i% | which expresses 



the simple movement of heat in the sphere. The number n has an 



infinity of values given by the definite equation - - = 1 - hX. The 



tan nX 



radius of the sphere is denoted by X, and the radius of any concentric 

 sphere, whose temperature is v after the lapse of the time t, by x\ h 

 and K are the specific coefficients; A is any constant. Constructions 

 adapted to disclose the nature of the definite equation, the limits and 



values of its roots 268 



290 292. Formation of the general solution ; final state of the solid . . 274 

 293. Application to the case in which the sphere has been heated by a pro 

 longed immersion ,..,.. 277 



SECTION n. 



DlFFEBENT BEMABKS ON THIS SOLUTION. 



294 296. Kesults relative to spheres of small radius, and to the final tem 

 peratures of any sphere ...... 279 



298300. Variable temperature of a thermometer plunged into a liquid 

 which is cooling freely. Application of the results to the comparison and 

 use of thermometers , , 282 



