242 
CALCULUS 
All this is plausible enough, but how do we know it is so, and 
what do we mean by heat, anyway ? To answer the second ques 
tion first, we think of heat as an imponderable substance which can 
flow freely in a conductor and which can be measured in calories,* 
as sugar in pounds. And now the above statements about heat, 
including equation (1) above and equation (2) below, are no more 
and no less than physical laics, — the facts of nature we take for 
granted. 
To go on: Consider a plane region S, of area A, situated in the 
slab and parallel to the faces. Let Q be the quantity of heat which 
traverses this surface in one second. Then obviously f Q is pro 
portional to A, to the difference in temperature of the faces, and in 
versely to the thickness of the plate: 
Q cc A, u x — u 0 , 
a 
(2) 
Q = K~ l ~ u *A, 
where K is a physical constant, the specific conductivity. 
Next, let the plane area S be oblique to the faces, making an 
angle 6 with them. Then the amount of heat, Q, which traverses 
the surface in one second will obviously be the same as that which 
traverses the projection, A cos 6, of S on a face, or 
(3) 
Q = K -l-~— 0 A cos 6. 
a 
The Normal Derivative. Consider the normal drawn to S in the 
sense of the flow. Let n be its length. Then it is readily seen 
that 
(4) 
du du r, 
— = — cos 6. 
on cx 
Lor 
XL p* —— LCptt y 
A u _ A u 
Am cos 6 Ax ' 
Ax = An cos 6, 
A u — Up' — Up = Up» — Up, 
and it remains merely to take the limits. 
* By a calorie is meant the quantity of heat required to raise one gramme of 
water one degree centigrade (the initial temperature being 15°). 
t Each statement is a physical law. “ Obviously ” means merely that these 
laws are easily accepted.