SECT. VI.] GENERAL EQUATIONS OF PROPAGATION. 109 



one side of this plane, exert on all the points /JL, //, /z&quot;, p&quot; , etc 

 situated on the other side : hence the constant flow, which, during 

 unit of time, crosses a definite part of the plane M in the infinite 

 solid, is equal to the quantity of heat which flows in the same time 

 across the same portion of the plane H in the prism, all of whose 

 temperatures are expressed by the equation 



v = A ax by - cz. 



138. COROLLARY. The flow has the value cK in the infinite 

 solid, when the part of the plane which it crosses has unit of 



surface. In the pi~ism also it has the same value cK or K -7- . 



It is proved in the same manner, that the constant flow which takes 

 place, during unit of time, in the. same prism across unit of surf ace t 

 on any plane whatever perpendicular to y, is equal to 



dv 

 bK or K 3- : 



&amp;lt;ty 



and that which crosses a plane perpendicular to x lias the value 



-. 



dx 



139. The propositions which we have proved in the preceding 

 articles apply also to the case in which the instantaneous action of 

 a molecule is exerted in the interior of the mass up to an appre 

 ciable distance. In this case, we must suppose that the cause 

 which maintains the external layers of the body in the state 

 expressed by the linear equation, affects the mass up to a finite 

 depth. All observation concurs to prove that in solids and liquids 

 the distance in question is extremely small. 



140. THEOREM III. If the temperatures at the points of a 

 solid are expressed by the equation v = f (x, y, z, t), in which 

 a?, y, z are the co-ordinates of a molecule whose temperature is 

 equal to v after the lapse of a time t; the flow of heat which 

 crosses part of a plane traced in the solid, perpendicular to one of 

 the three axes, is no longer constant ; its value is different for 

 different parts of the plane, and it varies also with the time. This 

 variable quantity may be determined by analysis. 



