SECT. VIII.] MOVEMENT IX A SOLID MASS. 81 



, , dv dv dv 

 w = v + -j + -T- *? + -j- ? 

 ax * dgp cfe 



will coincide as nearly as possible with the state of the solid ; that 

 is to say, all the molecules infinitely near to the point m will have 

 the same temperature, whether we consider them to be in the solid 

 or in the prism. This coincidence of the solid and the prism is 

 quite analogous to that of curved surfaces with the planes which 

 touch them. 



It is evident, from this, that the quantity of heat which flows 

 in the solid across the circle co, during the instant dt, is the same 

 as that which flows in the prism across the same circle; for all the 

 molecules whose actions concur in one effect or the other, have 

 the same temperature in the two solids. Hence, the flow in 



question, in one solid or the other, is expressed by K -=- wdt. 

 It would be K -=- codt, if the circle o&amp;gt;, whose centre is m, were 



perpendicular to the axis of y, and K -^- codt, if this circle were 

 perpendicular to the axis of x. 



The value of the flow which we have just determined varies 

 in the solid from one point to another, and it varies also with 

 the time. We might imagine it to have, at all the points of a 

 unit of surface, the same value as at the point m, and to preserve 

 this value during unit of time ; the flow would then be expressed 



by K-j- , it would be K-j- in the direction of y, and K~ 

 dz, dy dx 



in that of x. We shall ordinarily employ in calculation this 

 value of the flow thus referred to unit of time and to unit of 

 surface. 



99. This theorem serves in general to measure the velocity 

 with which heat tends to traverse a given point of a plane 

 situated in any manner whatever in the interior of a solid whose 

 temperatures vary with the time. Through the given point m, 

 a perpendicular must be raised upon the plane, and at every 

 point of this perpendicular ordinates must be drawn to represent 

 the actual temperatures at its different points. A plane curve 

 will thus be formed whose axis of abscissse is the perpendicular. 



F. H. 6 



