LINE .INTEGRALS AND GREEN’S THEOREM 245 
We have treated K as constant, and it is so for ordinary substances 
and moderate variations in temperature. The Hypothesis includes 
the case, however, that K is a continuous function of x, y, z, t. 
EXERCISE 
Show that the lines of flow are given by the simultaneous system 
of differential equations: 
dx _dy _ dz 
du du du 
dx dy dz 
13. A New Heat Problem. If a homogeneous substance be raised 
from the constant initial temperature u 0 to the constant final tem 
perature u x , the quantity of heat required, Q, will be proportional to 
the rise in temperature and the volume : 
Q oc Ui — u 0 , V; 
Q == C(u 1 -u 0 )V, 
• (9) 
where C is a physical constant depending on the substance, the 
specific heat per unit of volume. 
If, now, an arbitrary homogeneous substance be raised from the 
continuous initial temperature w 0 to the continuous final temperature 
?q, the amount of heat required will be 
(10) 
V 
as is shown by the usual procedure of the integral calculus. 
Consider an arbitrary flow of heat. Let the temperature, 
u=f(x, y, z, t) 
be Uq when t = t 0 and u x when t = + Ah The quantity of heat 
required to produce this change is given by (10), where for Q we 
now write AQ. 
On the other hand, 
U\ Uq — Atf(Xj y, z, t Q -f- 0 A/), 
0 < 6 < 1. 
Hence