CHAPTER VII. 



PROPAGATION OF HEAT IN A RECTANGULAR PRISM. 



321. THE equation ^ + ^4 + j^ = 0, which we have stated 



in Chapter II., Section iv., Article 125, expresses the uniform move 

 ment of heat in the interior of a prism of infinite length, sub 

 mitted at one end to a constant temperature, its initial tempera 

 tures being supposed nul. To integrate this equation we shall, 

 in the first place, investigate a particular value of v, remarking 

 that this function v must remain the same, when y changes sign 

 or when z changes sign ; an.d that its value must become infinitely 

 small, when the distance x is infinitely great. From this it is 

 easy to see that we can select as a particular value of v the 

 function ae~ mx cos ny cos pz ; and making the substitution we find 

 m z n 3 p z 0. Substituting for n and p any quantities what 

 ever, we have m = Jtf+p*. The value of v must also satisfy the 



definite equation I v + 2~ = ^ when y = l or ~Z, and the equation 



k V + ~dz = Wll6n Z = l r ~ l ( Cna pter II., Section IV., Article 125). 

 If we give to v the foregoing value, we have 



n sin ny + 7 cos ny = Q and p sin pz + 7 cospz = 0, 



hi hi 



or -j- = pi tan pi, -r = nl tan nl. 



We see by this that if we find an arc e, such that etane is equal 

 to the whole known quantity T I, we can take for n or p the quan- 



