92 THEORY OF HEAT. [CHAP. II. 



heat which accumulates in the intervening shell, and whose effect 

 is to vary its temperature. 



113. The coefficient C denotes the quantity of heat which is 

 necessary to raise, from temperature to temperature 1, a definite 

 unit of weight ; D is the weight of unit of volume, ^Trx^dx is the 

 volume of the intervening layer, differing from it only by a 

 quantity which may be omitted : hence kjrCDx^dx is the quantity 

 of heat necessary to raise the intervening shell from temperature 

 to temperature 1. Hence it is requisite to divide the quantity 

 of heat which accumulates in this shell by 4 r jrCDx 2 dx ) and we 

 shall then find the increase of its temperature v during the time 

 dt. We thus obtain the equation 



Jr d(x 2 } 

 , _ K , \ dxj 



~ CD x*dx 



v 2 dv\ 



or -77 = TTT: I -r-a + - -7- / (c). 



5 x dxj ^ 



114. The preceding equation represents the law of the move 

 ment of heat in the interior of the solid, but the temperatures of 

 points in the surface are subject also to a special condition which 

 must be expressed. This condition relative to the state of the 

 surface may vary according to the nature of the problems dis 

 cussed : we may suppose for example, that, after having heated 

 the sphere, and raised all its molecules to the temperature of 

 boiling water, the cooling is effected by giving to all points in the 

 surface the temperature 0, and by retaining them at this tem 

 perature by any external cause whatever. In this case we may 

 imagine the sphere, whose variable state it is desired to determine, 

 to be covered by a very thin envelope on which the cooling agency 

 exerts its action. It may be supposed, 1, that this infinitely 

 thin envelope adheres to the solid, that it is of the same substance 

 as the solid and that it forms a part of it, like the other portions 

 of the mass ; 2, that all the molecules of the envelope are sub 

 jected to temperature Oby a cause always in action which prevents 

 the temperature from ever being above or below zero. To express 

 this condition theoretically, the function v, which contains x and t, 



