SECT. VII.] GENERAL SURFACE EQUATION. 121 



152. One of the faces perpendicular to x is a triangle, and 

 the opposite face is a trapezium. The area of the triangle is 



1 , ch 



and the flow of heat in the direction perpendicular to this surface 

 -y- 



CLOO 



being K -y- we have, omitting the factor dt, 



dz 



as the expression of the quantity of heat which in one instant 

 passes into the molecule, across the triangle in question. 

 The area of the opposite face is 



1 j f dz , , dz , dz , \ 

 - ay [ -j ax + -y- ax + -j~ ay , 



2 9 \dx dx dy y j 



CM ?7 



and the flow perpendicular to this face is also K-J-, suppress 

 ing terms of the second order infinitely smaller than those of the 

 first; subtracting the quantity of heat which escapes by the second 

 face from that which enters by the first we find 



T rdv dz j j 



K -7- -j- dx dy. 

 dx dx 



This term expresses the quantity of heat the molecule receives 

 through the faces perpendicular to x. 



It will be found, by a similar process, that the same molecule 

 receives, through the faces perpendicular to y, a quantity of heat 



, , vr dv dz , , 

 equal to K -^ j dx dy. 



The quantity of heat which the molecule receives through the 



dv 



rectangular base is K-j-dx dy. Lastly, across the upper sur 

 face a Vc d , a certain quantity of heat is permitted to escape, 

 equal to the product of hv into the extent co of that surface. 

 The value of o&amp;gt; is, according to known principles, the same as that 



of dx dy multiplied by the ratio - ; e denoting the length of the 

 normal between the external surface and the plane of x and ?/, and 



fdz\* (dz 



4- l-T- + (- 



j \dy 



