SECT. I.] VARIED MOVEMENT IN A RING. 8? 



which escapes from the same slice across the second section, and 

 passes into the contiguous part of the solid, it is only necessary 

 to change x into x 4- dx in the preceding expression, or, which is 

 the same thing, to add to this expression its differential taken 

 with respect to x ; thus the slice receives through one of its faces 



a quantity of heat equal to KS-j-dt, and loses through the 

 opposite face a quantity of heat expressed by 



Tr . ~ - -, -rr- n , , 



- KS-j- dt - KS T-O dx dt. 

 dx dx 



It acquires therefore by reason of its position a quantity of heat 

 equal to the difference of the two preceding quantities, that is 



KSldxdt. 

 dx? 



On the other hand, the same slice, whose external surface is 

 Idx and whose temperature differs infinitely little from v, allows 

 a quantity of heat equivalent to hlvdxdt to escape into the air; 

 during the instant dt\ it follows from this that this infinitely- 

 small part of the solid retains in reality a quantity of heat 



72 



represented by K S -^ dx dt - hlv dx dt which makes its tempe- 

 clx 



rature vary. The amount of this change must be examined. 



105. The coefficient C expresses how much heat is required 

 to raise unit of weight of the substance in question from tempe 

 rature up to temperature 1 ; consequently, multiplying the 

 volume Sdx of the infinitely small slice by the density Z&amp;gt;, to 

 obtain its weight, and by C the specific capacity for heat, we shall 

 have CD Sdx as the quantity of heat which would raise the 

 volume of the slice from temperature up to temperature 1. 

 Hence the increase of temperature which results from the addition 



J7 



of a quantity of heat equal to KS -^ dx dt hlv dx dt will be 



found by dividing the last quantity by CD Sdx. Denoting there 

 fore, according to custom, the increase of temperature which takes 



place during the instant dt by -, y dt, we shall have the equation 



