CHAPTER II. 



EQUATIONS OF THE MOVEMENT OF HEAT. 



SECTION I. 

 Equation of the varied movement of heat in a ring. 



101. WE might form the general equations which represent 

 the movement of heat in solid bodies of any form whatever, and 

 apply them -to particular cases. But this method would often 

 involve very complicated calculations which may easily be avoided. 

 There are several problems which it is preferable to treat in a 

 special manner by expressing the conditions which are appropriate 

 to them; we proceed to adopt this course and examine separately 

 the problems which have been enunciated in the first section of 

 the introduction ; we will limit ourselves at first to forming the 

 differential equations, and shall give the integrals of them in the 

 following chapters. 



102. We have already considered the uniform movement of 

 heat in a prismatic bar of small thickness whose extremity is 

 immersed in a constant source of heat. This first case offered no 

 difficulties, since there was no reference except to the permanent 

 state of the temperatures, and the equation which expresses them 

 is easily integrated. The following problem requires a more pro 

 found investigation; its object is to determine the variable state 

 of a solid ring whose different points have received initial tempe 

 ratures entirely arbitrary. 



The solid ring or armlet is generated by the revolution of 

 a rectangular section about an axis perpendicular to the plane of 



