SECT. VIII.] MOVEMENT IN A CUBE. 83 



of the preceding article, that the value of the flow, at this point 



and across the plane, is K -j- , or Ke~ 3t cos x . cos y . sin z. The 



clz 



quantity of heat which, during the instant dt, crosses an infinitely 

 small rectangle, situated on this plane, and whose sides are 

 dx and dy, is 



K e* cos x cos y sin z dx dy dt. 



Thus the whole heat which, during the instant dt, crosses the 

 entire area of the same plane, is 



K e gf sin z . dt / / cos x cos ydxdy; 



the double integral being taken from x = ^ IT up to x = = TT, 



and from y = - TT up to y = - TT. We find then for the ex- 



* 



pression of this total heat, 



4 A V sin^.ok 



If then we take the integral with respect to t, from t = to 

 t = , we shall find the quantity of heat which has crossed the 

 same plane since the cooling began up to the actual moment. 



This integral is sin z (1 e~ gt ), its value at the surface is 



so that after an infinite time the quantity of heat lost through 

 one of the faces is . The same reasoning being applicable 

 to each of the six faces, we conclude that the solid has lost by its 



complete cooling a total quantity of heat equal to - - or SCD, 



*J 



since g is equivalent to -^^ . The total heat which is dissipated 



C.L/ 



during the cooling must indeed be independent of the special 

 conducibility K, which can only influence more or less the 

 velocity of cooling. 



C 2 



