SECT. VIII.] GENERAL EQUATIONS APPLIED. 123 



enter into the expression of the heat acquired cancel each other ; 

 so that the gain of heat cannot be expressed except by terms 

 of the second order. We may give to the molecule the form, 

 either of a right prism whose axis is normal to the surface of the 

 solid, or that of a truncated prism, or any form whatever. 



The general equation (A), (Art. 142) supposes that all the 

 terms of the first order cancel each other in the interior of the 

 mass, which is evident for prismatic molecules enclosed in the 

 solid. The equation (B), (Art. 147) expresses the same result 

 for molecules situated at the boundaries of bodies. 



Such are the general points of view -from which we may look 

 at this part of the theory of heat. 



, dv K fd*v d*v &amp;lt;Fv\ ,, 



The equation ^ = m (^ + jf+&) represents the move- 



ment of heat in the interior of bodies. It enables us to ascer 

 tain the distribution from instant to instant in all substances 

 solid or liquid ; from it we may derive the equation which 

 belongs to each particular case. 



In the two following articles we shall make this application 

 to the problem of the cylinder, and to that of the sphere. 



SECTION VIII. 



Application of the general equations. 



155. Let us denote the variable radius of any cylindrical 

 envelope by r, and suppose, as formerly, in Article 118, that 

 all the molecules equally distant from the axis have at each 

 instant a common temperature ; v will be a function of r and t ; 

 r is a function of y, z, given by the equation r 2 = y z + z*. It is 

 evident in the first place that the variation of v with respect 



73 



to x is nul : thus the term -j-s must be omitted. We shall have 



dx* 



then, according to the principles of the differential calculus, the 

 equations 



dv_dvdr , d*v _ d?v_ (dr\* dv 

 Ty ~ dr Ty J ~df~dr* [dy) + d 



dv dv dr , d 2 v 



~r~ = i r aud ~ra 

 dz dr dz dz z 



d*v fdr\* dv fd*r\ 



= ~rr I ~5~ I + T~ I -i~ ; 



dr* \dz) dr \dz*J 



