SECT. IV.] FINAL PERMANENT STATE. 1G1 



199. By this it may be seen how the analysis satisfies the 

 two conditions of the hypothesis, which subjected the base to a 

 temperature equal to cosy, and the two sides B and C to the 

 temperature 0. When we express these t\vo conditions we solve 

 in fact the following problem : If the heated plate formed part of 

 an infinite plane, what must be the temperatures at all the points 

 of the plane, in order that the system may be self-permanent, and 

 that the fixed temperatures of the infinite rectangle may be those 

 which are given by the hypothesis ? 



We have supposed in the foregoing part that some external 

 causes maintained the faces of the rectangular solid, one at the 

 temperature 1, and the two others at the temperature 0. This 

 effect may be represented in different manners; but the hypo 

 thesis proper to the investigation consists in regarding the prism 

 as part of a solid all of whose dimensions are infinite, and in deter 

 mining the temperatures of the mass which surrounds it, so that 

 the conditions relative to the surface may be always observed. 



200. To ascertain the system of permanent temperatures in 

 a rectangular plate whose extremity A is maintained at the tem 

 perature 1, and the two infinite edges at the temperature 0, we 

 might consider the changes which the temperatures undergo, 

 from the initial state which is given, to the fixed state which is 

 the object of the problem. Thus the variable state of the solid 

 would be determined for all values of the time, and it might then 

 be supposed that the value was infinite. 



The method which we have followed is different, and conducts 

 more directly to the expression of the final state, since it is 

 founded on a distinctive property of that state. We now proceed 

 to shew that the problem admits of no other solution than that 

 which we have stated. The proof follows from the following 

 propositions. 



201. If we give to all the points of an infinite rectangular 

 plate temperatures expressed by equation (2), and if at the two 

 edges B and C we maintain the fixed temperature 0, whilst the 

 end A is exposed to a source of heat which keeps all points of the 

 line A at the fixed temperature 1; no change can happen in the 



state of the solid. In fact, the equation -y- a + -=-$ = being 



F. H. n 



