CHAPTER VIII. 



OF THE MOVEMENT OF HEAT IN A SOLID CUBE. 



333. IT still remains for us to make use of the equation 

 dv K /d?v d*v a 



which represents the movement of heat in a solid cube exposed 

 to the action .of the air (Chapter II., Section v.). Assuming, in 

 the first place, for v the very simple value e~ mt cosnx cospycosqz, 

 if we substitute it in the proposed equation, we have the equa 

 tion of condition m = k (n* + p* + q*), the letter k denoting the 



TT- 



coefficient . It follows from this that if we substitute for 



n, p, q any quantities whatever, and take for m the quantity 

 k(n z + p* + q 2 ), the preceding value of v will always satisfy the 

 partial differential equation. We have therefore the equation 

 v = e - k (n*+ P * + q 2 )t cos nx cospycosqz. The nature of the problem 

 requires also that if x changes sign, and if y and z remain the 

 same, the function should not change ; and that this should also 

 hold with respect to y or z: now the value of v evidently satisfies 

 these conditions. 



334. To express the state of the surface, we must employ the 

 following equations : 



.(6). 



212 



