CHAPTER III. 







PROPAGATION OF HEAT IN AN INFINITE RECTANGULAR SOLID. 



SECTION I. 



Statement of the problem. 



163. PROBLEMS relative to the uniform propagation, or to 

 the varied movement of heat in the interior of solids, are reduced, 

 by the foregoing methods, to problems of pure analysis, and 

 the progress of this part of physics will depend in consequence 

 upon the advance which may be made in the art of analysis. 

 The differential equations which we have proved contain the 

 chief results of the theory ; they express, in the most general 

 and most concise manner, the necessary relations of numerical 

 analysis to a very extensive class of phenomena; and they 

 connect for ever with mathematical science one of the most 

 important branches of natural philosophy. 



It remains now to discover the proper treatment of these 

 equations in order to derive their complete solutions and an 

 easy application of them. The following problem offers the 

 first example of analysis which leads to such solutions ; it 

 appeared to us better adapted than any other to indicate the 

 elements of the method which we have followed. 



164. Suppose a homogeneous solid mass to be contained 

 between two planes B and G vertical, parallel, and infinite, and 

 to be divided into two parts by a plane A perpendicular to the 

 other two (fig. 7) ; we proceed to consider the temperatures of 

 the mass BAC bounded by the three infinite planes A t B, C. 

 The other part B AC of the infinite solid is supposed to be a 

 constant source of heat, that is to say, all its points are main 

 tained at the temperature 1, which cannot alter. The two 



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