98 THEORY OF HEAT. [CHAP. II. 



122. The section made at right angles to the length of the 

 bar, is a square whose side is 2f, the axis of the bar is the axis 

 of x, and the origin is at the extremity A. The three rectangular 

 co-ordinates of a point of the bar are x t y, z, and v denotes the 

 fixed temperature at the same point. 



The problem consists in determining the temperatures which 

 must be assigned to different points of the bar, in order that 

 they may continue to exist without any change, so long as the 

 extreme surface A, which communicates with the source of heat, 

 remains subject, at all its points, to the permanent tempera 

 ture A ; thus v is a function of x t y, and z. 



123. Consider the movement of heat in a prismatic molecule, 

 enclosed between six planes perpendicular to the three axes 

 of x, y, and z. The first three planes pass through the point m 

 whose co-ordinates are x, y, z, and the others pass through the 

 point m whose co-ordinates are x -f dx, y + dy, z-\- dz. 



To find what quantity of heat enters the molecule during 

 unit of time across the first plane passing through the point m 

 and perpendicular to x t we must remember that the extent of the 

 surface of the molecule on this plane is dydz, and that the flow 

 across this area is, according to the theorem of Article 98, equal 



to K ; thus the molecule receives across the rectangle dydz 

 dx 



passing through the point m a quantity of heat expressed by 

 z -j- . To find the quantity of heat which crosses the 



opposite face, and escapes from the molecule, we must substitute, 

 in the preceding expression, x + dx for x, or, which is the same 

 thing, add to this expression its differential taken with respect 

 to x only; whence we conclude that the molecule loses, at its 

 second face perpendicular to x, a quantity of heat equal to 



dv fdv\ 



A dydz , -- A dndzd -r- ; 

 9 dx \dxj 



we must therefore subtract this from that which enters at the 

 opposite face ; the differences of these two quantities is 



tr j j j fdv\ 



A dydz a I -j- 1 , or, A a x dyd 

 \ctx/ 



d 2 v 

 z -=-^ 

 dx 



