SECT. IV.] THE FINAL STATE IS UNIQUE. 165 



If this state were formed at first it would be self-existent, and 

 it is this property which has served to determine it for us. If the 

 solid plate be supposed to be in another initial state, the differ 

 ence between the latter state and the fixed state forms a partial 

 state, which imperceptibly disappears. After a considerable time, 

 the difference has nearly vanished, and the system of fixed tem 

 peratures has undergone no change. Thus the variable temper 

 atures converge more and more to a final state, independent of 

 the primitive heating. 



204. We perceive by this that the final state is unique; for, 

 if a second state were conceived, the difference between the 

 second and the first would form a partial state, which ought to be 

 self-existent, although the edges A, B, C were maintained at the 

 temperature 0. Now the last effect cannot occur; similarly if we 

 supposed another source of heat independent of that which flows 

 from the origin A] besides, this hypothesis is not that of the 

 problem we. have treated, in which the initial temperatures are 

 nul. It is evident that parts very distant from the origin can 

 only acquire an exceedingly small temperature. 



Since the final state which must be determined is unique, it 

 follows that the problem proposed admits no other solution than 

 that which results from equation (a). Another form may be 

 given to this result, but the solution can be neither extended nor 

 restricted without rendering it inexact. 



The method which we have explained in this chapter consists 

 in formnig fiFst very simple particular values, which agree with 

 the .problem, and in rendering the solution more general, to the 

 intent that v or &amp;lt;/&amp;gt; (as, y) may satisfy three conditions, namely : 



It is clear that the contrary order might be followed, and the 

 solution obtained would necessarily be the same as the foregoing. 

 We shall not stop over the details, which are easily supplied, 

 when once the solution is known. We shall only give in the fol 

 lowing section a remarkable expression for the function &amp;lt;/&amp;gt; (x, y] 

 whose value was developecTm a convergent series in equation (a). 



