SECT. IV.] STEADY MOVEMENT IN A PRISM. 9D 



this expresses the quantity of heat accumulated in the molecule 

 in consequence of the propagation in direction of x ; which ac 

 cumulated heat would make the temperature of the molecule 

 vary, if it were not balanced by that which is lost in some other 

 direction. 



It is found in the same manner that a quantity of heat equal 



to Kdz dx -T- enters the molecule across the plane passing 



through the point m perpendicular to y, and that the quantity 

 which escapes at the opposite face is 



Kdzdx -j -- Kdzdx d ( - T - ) , 

 dy \dy) 



the last differential being taken with respect to y only. Hence 



the difference of the two quantities, or Kdxdydz j-$, expresses 



dy 



the quantity of heat which the molecule acquires, in consequence 

 of the propagation in direction of y. 



Lastly, it is proved in the same manner that the molecule 

 acquires, in consequence of the propagation in direction of z t 



a quantity of heat equal to Kdxdydz-j-j. Now, in order that 



dz 



there may be no change of temperature, it is necessary for the 

 molecule to retain as much heat as it contained at first, so that 

 the heat it acquires in one direction must baknce that 

 loses in another. Hence the sum of the three quanti 

 acquired must be nothing; thus we form the equation 



d 2 v cPv tfv _ 

 da?d** dz z ~ 



first, so that 

 hat which it 

 ities of heat 



124 It remains now to express the conditions relative to the 

 surface. If we suppose the point m to belong to one of the faces 

 of the prismatic bar, and the face to be perpendicular to z, we 

 see that the rectangle dxdy, during unit of time, permits a 

 quantity of heat equal to Vh dx dy to escape into the air, 

 V denoting the temperature of the point m of the surface, namely 

 what &amp;lt;f&amp;gt; (x, y, z] the function sought becomes when z is made 

 equal to I, half the dimension of the prism. On the other hand, 

 the quantity of heat which, by virtue of the action of the 



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