LINE INTEGRALS AND GREEN’S THEOREM 243 
From (1) we find : 
du 
cx 
U y — V-o 
a 
On combining this equation with (3) and eliminating du/dx by 
means of (4), we obtain: 
Q = -K d ^A. 
cn 
(5) 
This result embodies all the physical laws that have gone before, 
for from it we can deduce both (1) and (3). Moreover, it states 
these laws in terms of what is going on in the neighborhood of P, 
and not in terms of the temperature at remote points. 
If the sense of the normal be reversed, Q will be replaced by its 
negative. 
12. Continuation. The General Case. Consider now an arbitrary 
steady flow. The lines of flow will be curved lines, forming a two- 
parameter family of space curves which just fill out the region of 
flow. These curves are obtained by considering the family of 
isothermal surfaces, 
(6) u = ll 0 . 
In the neighborhood of any point P within the region of flow, the 
situation is similar to that set forth in § 11, for the portions of these 
surfaces contained in the neighborhood of P look almost like planes, 
which are sensibly parallel to one another, and so the lines of flow 
are seen to be curves cutting these surfaces orthogonally. 
Let N be a surface, open or closed, which lies in the region of 
flow. CutN up into n pieces ASy, •••, AS n , the maximum diameters 
of these pieces being small. Then each of the pieces will look like a 
small piece of a plane surface, and it is physically evident that t'he 
amount of heat which traverses the &-tli region in one second will be 
approximately 
where the normal derivative is formed at an arbitrary point of that 
region. Thus the quantity, Q, of heat which traverses the whole 
surface S in one second will be approximately 
and the approximation will be closer and closer, the smaller the sub- 
regions, or