SECT. VII.] GENEKAL SURFACE EQUATION. 119 



7 rs 



The second term K -r- co ~- dt represents the quantity of 



heat which would cross the projection of a), made on the plane of 

 x and z, if this projection were placed parallel to itself at the 

 point p. 



7 rj 



Lastly, the third term - K -j- co -~-dt represents the quantity 



of heat which would flow during the instant dt, across the projec 

 tion of o&amp;gt; on the plane of so and y, if this projection were placed at 

 the point p, perpendicular to the co-ordinate z. 



By this it is seen that the quantity of heat which flows across 

 every infinitely small part of a surface drawn in the interior of the 

 solid, can always be decomposed into three other quantities of flow, 

 which penetrate the three orthogonal projections of the surface, along 

 the directions perpendicular to the planes of the projections. The 

 result gives rise to properties analogous to those which have 

 been noticed in the theory of forces. 



150. The quantity of heat which flows across a plane surface 

 ft&amp;gt;, infinitely small, given in form and position, being equivalent 

 to that which would cross its three orthogonal projections, it fol 

 lows that, if in the interior of the solid an element be imagined of 

 any form whatever, the quantities of heat which pass into this 

 polyhedron by its different faces, compensate each other recipro 

 cally: or more exactly, the sum of the terms of the first order, 

 which enter into the expression of the quantities of heat received 

 by the molecule, is zero ; so that the heat which is in fact accumu 

 lated in it, and makes its temperature vary, cannot be expressed 

 except by terms infinitely smaller than those of the first order. 



This result is distinctly seen when the general equation (A) 

 has been established, by considering the movement of heat in 

 a prismatic molecule (Articles 127 and 142) ; the demonstration 

 may be extended to a molecule of any form whatever, by sub 

 stituting for the heat received through each face, that which its 

 three projections would receive. 



In other respects it is necessary that this should be so : for, if 

 one of the molecules of the solid acquired during each instant a 

 quantity of heat expressed by a term of the first order, the varia 

 tion of its temperature would be infinitely greater than that of 



