SECT. V.] VARIED MOVEMENT IN A CUBE 8 . 10 T 



I or I, whatever x and z may be, or satisfies* the equation 

 -pV + ^- = 0, when z is equal to I or I, whatever x and y may 



be ; 3rd, it satisfies the equation v = A, when x = 0, whatever 

 y and z may be. 



SECTION Y. 

 Equations of the varied movement of heat in a solid cule. 



126. A solid in the form of a cube, all of whose points have 

 acquired the same temperature, is placed in a uniform current of 

 atmospheric air, maintained at temperature 0. It is required to 

 determine the successive states of the body during the whole 

 time of the cooling. 



The centre of the cube is taken as the origin of rectangular 

 coordinates; the three perpendiculars dropped from this point on 

 the faces, are the axes of x, y, and z ; 21 is the side of the cube, 

 v is the temperature to which a point whose coordinates are 

 x, y } z, is lowered after the time t has elapsed since the com 

 mencement of the cooling : the problem consists in determining 

 the function v, which depends on x, y, z and t. 



127. To form the general equation which v must satisfy, 

 we must ascertain what change of temperature an infinitely 

 small portion of the solid must experience during the instant 

 dt, by virtue of the action of the molecules which are extremely 

 near to it. We consider then a prismatic molecule enclosed 

 between six planes at right angles; the first three pass through 

 the point m, whose co-ordinates are x, y, z, and the three others, 

 through the point m , whose co-ordinates are 



x + dx, y + dy, z + dz. 



The quantity of heat which during the instant dt passes into 

 the molecule across the first rectangle dy dz perpendicular to x, 



is Kdy dz -T- dt, and that which escapes in the same time from 



the molecule, through the opposite face, is found by writing 

 x-}- dx in place of x in the preceding expression, it is 



- Kdy ^ ( -y-J dt. Kdy dzd(-^\ dt, 



