126 THEORY OF HEAT. [CHAP. II. 



The three equations of the second order give : 



dr\* 

 dy *V dx z dy* 



and substituting for 



_(dr\ fdr\ 



&quot; \dx) + \dy) + T z + * + 



dx 

 its value 1, we have 



ffr 



Making these substitutions in the equation (a) we have the 

 equation 



~dt^UD |&amp;lt;F + r ~&amp;lt;FJ 



which is the same as that of Art. 114. 



The equation would contain a greater number of terms, if we 

 supposed molecules equally distant from the centre not to have 

 received the same initial temperature. 



We might also deduce from the definite equation (B), the 

 equations which express the state of the surface in particular 

 cases, in which we suppose solids of given form to communicate 

 their heat to the atmospheric air ; but in most cases these equa 

 tions present themselves at once, and their form is very simple, 

 when the co-ordinates are suitably chosen. 



SECTION IX. 

 General Remarks. 



157. The investigation of the laws of movement of heat in 

 solids now consists in the integration of the equations which we 

 have constructed ; this is the object of the following chapters. 

 We conclude this chapter with general remarks on the nature 

 of the quantities which enter into our analysis. 



In order to measure these quantities and express them nume 

 rically, they must be compared with different kinds of units, five 



