100 THEORY OF HEAT. [CHAP. II. 



molecules, during unit of time, traverses an infinitely small surface 

 G&amp;gt;, situated within the prism, perpendicular to z y is equal to 



Kco-j-, according to the theorems quoted above. This ex- 



pression is general, and applying it to points for which the co 

 ordinate z has its complete value I, we conclude from it that the 

 quantity of heat which traverses the rectangle dx dy taken at the 



surface is - Kdxdy-j-, giving to z in the function -7- its com 

 plete value I. Hence the two quantities Kdxdy-j-, and 



CLZ 



h dx dy v, must be equal, in order that the action of the molecules 

 may agree with that of the medium. This equality must also 



exist when we give to z in the functions -y- and v the value I, 



dz 



which it has at the face opposite to that first considered. Further, 

 the quantity of heat which crosses an infinitely small surface co, 



perpendicular to the axis of y, being Kco-j-, it follows that 

 that which flows across a rectangle dz dx taken on a face of the 



(i rJ 



prism perpendicular to y is - K dz dx -=- , giving to y in the 



J 



function -y- its complete value I. Now this rectangle dz dx 

 dy 



permits a quantity of heat expressed by hv dx dy to escape into 

 the air; the equation hv = K^- becomes therefore necessary, 



t/ 



r/?j 



when y is made equal to I or I in the functions v and -=- . 



dy 



125. The value of the function v must by hypothesis be 

 equal to A, when we suppose a? = 0, whatever be the values of 

 y and z. Thus the required function v is determined by the 

 following conditions: 1st, for all values of x } y, z, it satisfies the 

 general equation 



d^v d*v d*v _ 



dtf + dy* + ~dz*~ 



2nd, it satisfies the equation y^w + -r- = 0, when y is equal to 



