TABLE OF CONTENTS. 



SECTION VI. 



GENERAL EQUATION OF THE PROPAGATION OF HEAT IN THE INTERIOR 

 OF SOLIDS. 



ART. PAGE 



132139. Elementary proof of properties of the uniform movement of heat 

 in a solid enclosed between six orthogonal planes, the constant tem 

 peratures being expressed by the linear equation, 



v = A - ax - by - cz. 



The temperatures cannot change, since each point of the solid receives 

 as much heat as it gives off. The quantity of heat which during the 

 unit of time crosses a plane at right angles to the axis of z is the same, 

 through whatever point of that axis the plane passes. The value of this 

 common flow is that which would exist, if the coefficients a and 6 



were nul 104 



140, 141. Analytical expression of the flow in the interior of any solid. The 



equation of the temperatures being v=f(x, y, z, t) the function -Ku 



expresses the quantity of heat which during the instant dt crosses an 

 infinitely small area w perpendicular to the axis of z, at the point whose 

 coordinates are x, ?/, z, and whose temperature is v after the time t 



has elapsed 109 



142 145. It is easy to derive from the foregoing theorem the general 

 equation of the movement of heat, namely 



dv K 



SECTION VII. 



GENERAL EQUATION BELATIVE TO THE SURFACE. 



146 154. It is proved that the variable temperatures at points on the 

 surface of a body, which is cooling in air, satisfy the equation 



dv dv dv h 



being the differential equation of the surface which bounds the solid, 

 and q being equal to (m? + n*+p *)2. To discover this equation we 

 consider a molecule of the envelop which bounds the solid, and we express 

 the fact that the temperature of this element does not change by a finite 

 magnitude during an infinitely small instant. This condition holds and 

 continues to exist after that the regular action of the medium has been 

 exerted during a very small instant. Any form may be given to the 

 element of the envelop. The case in which the molecule is formed by 

 rectangular sections presents remarkable properties. In the most simple 

 case, which is that in which the base is parallel to the tangent plane, 

 the truth of the equation is evident ..... 115 



