CH. I. SECT. I.] INTRODUCTION. 15 



determining what is the temperature at each point of a body 

 at a given instant, supposing that the initial temperatures are 

 known. The following examples will more clearly make known 

 the nature of these problems. 



2. If we expose to the continued and uniform action of a 

 source of heat, the same part of a metallic ring, whose diameter 

 is large, the molecules nearest to the source will be first heated, 

 and, after a certain time, every point of the solid will have 

 acquired very nearly the highest temperature which it can attain. 

 This limit or greatest temperature is not the same at different 

 points ; it becomes less and less according as they become more 

 distant from that point at which the source of heat is directly 

 applied. 



When the temperatures have become permanent, the source 

 of heat supplies, at each instant, a quantity of heat which exactly 

 compensates for that which is dissipated at all the points of the 

 external surface of the ring. 



If now the source be suppressed, heat will continue to be 

 propagated in the interior of the solid, but that which is lost 

 in the medium or the void, will no longer be compensated as 

 formerly by the supply from the source, so that all the tempe 

 ratures will vary and diminish incessantly until they have be 

 come equal to the temperatures of the surrounding medium. 



3. Whilst the temperatures are permanent and the source 

 remains, if at every point of the mean circumference of the ring 

 an ordinate be raised perpendicular to the plane of the ring, 

 whose length is proportional to the fixed temperature at that 

 point, the curved line which passes through the ends of these 

 ordi nates will represent the permanent state of the temperatures, 

 and it is very easy to determine by analysis the nature of this 

 line. It is to be remarked that the thickness of the ring is 

 supposed to be sufficiently small for the temperature to be 

 sensibly equal at all points of the same section perpendicular 

 to the mean circumference. When the source is removed, the 

 line which bounds the ordinates proportional to the temperatures 

 at the different points will change its form continually. The 

 problem consists in expressing, by one equation, the variable 



