SECT. VII.] GENERAL SURFACE EQUATION. 115 



fusion of heat, and the law to which this principle is submitted 

 becomes more manifest ; it is expressed by the general equation 



dt ~ CD 



to which must be added that which relates to the initial arbitrary 

 state of the solid. 



Suppose the initial temperature of a molecule, whose co 

 ordinates are x, y, z } to be a known function F(x t y, z} y and denote 

 the unknown value v by &amp;lt;f&amp;gt; (x, y, z, t), we shall have the definite 

 equation &amp;lt;f&amp;gt; (as, y, z, 0) = F (x, y, 2) ; thus the problem is reduced to 

 the integration of the general equation (A) in such a manner that 

 it may agree, when the time is zero, with the equation which con 

 tains the arbitrary function F. 



SECTION VII. 



General equation relative to the surface. 



146. If the solid has a definite form, and if its original heat 

 is dispersed gradually into atmospheric air maintained at a con 

 stant temperature, a third condition relative to the state of the 

 surface must be added to the general equation (A) and to that 

 which represents the initial state. 



We proceed to examine, in the following articles, the nature of 

 the equation which expresses this third condition. 



Consider the variable state of a solid whose heat is dispersed 

 into air, maintained at the fixed temperature 0. Let o&amp;gt; be an 

 infinitely small part of the external surface, and p a point of &&amp;gt;, 

 through which a normal to the surface is drawn ; different points 

 of this line have at the same instant different temperatures. 



Let v be the actual temperature of the point p,, taken at a 

 definite instant, and w the corresponding temperature of a point v 

 of the solid taken on the normal, and distant from //, by an in 

 finitely small quantity a. Denote by x, y, z the co-ordinates of 

 the point p, and those of the point v by x + &, y + &y, z + Sz ; 

 let/ (x, y, z) = be the known equation to the surface of the solid, 

 and v = &amp;lt;/&amp;gt; (x, y, z, f) the general equation which ought to give the 



82 



