TABLE OF CONTENTS. XV11 



ART. PAGB 



301. Expression of the mean temperature of the sphere as a function of the 



time elapsed 286 



302 304. Application to spheres of very great radius, and to those in which 



the radius is very small 287 



305. Kernark on the nature of the definite equation which gives all the values 



of n . ,289 



CHAPTER VI. 



Of the Movement of Heat in a solid cylinder. 



306, 307. We remark in the first place that the ratio of the variable tem 

 peratures of two points of the solid approaches continually a definite 

 limit, and by this we ascertain the expression of the simple movement. 

 The function of x which is one of the factors of this expression is given 

 by a differential equation of the second order. A number g enters into 

 this function, and must satisfy a definite equation 291 



308, 309. Analysis of this equation. By means of the principal theorems of 



algebra, it is proved that all the roots of the equation are real . . . 294 



310. The function u of the variable x is expressed by 



i r 1 * i 



u = / dr cos (xtjg sin r) ; 



and the definite equation is hu + =0, giving to x its complete value X. 296 

 311, 312. The development of the function $(z) being represented by 



2 2 , 2 



&quot; f&C&amp;gt; 



the value of the series 



c&amp;lt; 2 et* 



2 2 2 2 . 4 2 2 2 . 4 2 . 6 2 



1 t* 



is / dii(f&amp;gt;(tsmu). 



irJ Q 



Remark on this use of definite integrals ....... 298 



313. Expression of the function u of the variable a; as a continued fraction . 300 



314. Formation of the general solution 301 



315 318. Statement of the analysis which determines the values of the co 

 efficients 303 



319. General solution 308 



320. Consequences of the solution . . 309 



