86 THEOKY OF HEAT. [CHAP. II. 



the ring (see figure 3), I is the perimeter of the section whose area 

 * s ^ tne coen&amp;lt; i c i en t h measures the external con- 

 ducibility, K the internal conducibility, the 

 specific capacity for heat, D the density. The line 

 oxos x&quot; represents the mean circumference of the 

 armlet, or that line which passes through the 

 centres of figure of all the sections; the distance 

 of a section from the origin o is measured by the 



arc whose length is x\ R is the radius of the mean circumference. 

 It is supposed that on account of the small dimensions and of 



the form of the section, we may consider the temperature at the 



different points of the same section to be equal. 



103. Imagine that initial arbitrary temperatures have been 

 given to the different sections of the armlet, and that the solid is 

 then exposed to air maintained at the temperature 0, and dis 

 placed with a constant velocity; the system of temperatures will 

 continually vary, heat will be propagated within the ring, and 

 dispersed at the surface: it is required to determine what will be 

 the state of the solid at any given instant. 



Let v be the temperature which the section situated at distance 

 x will have acquired after a lapse of time t ; v is a certain function 

 of x and t, into which all the initial temperatures also must enter : 

 this is the function which is to be discovered. 



104. We will consider the movement of heat in an infinitely 

 small slice, enclosed between a section made at distance x and 

 another section made at distance x -f dx. The state of this slice 

 for the duration of one instant is that of an infinite solid termi 

 nated by two parallel planes maintained at unequal temperatures ; 

 thus the quantity of heat which flows during this instant dt across 

 the first section, and passes in this way from the part of the solid 

 which precedes the slice into the slice itself, is measured according 

 to the principles established in the introduction, by the product of 

 four factors, that is to say, the conducibility K, the area of the 



section S, the ratio -=- , and the duration of the instant; its 

 dx 



expression is KS -j- dt. To determine the quantity of heat 



