244 
CALCULUS 
This whole deduction has been heuristic. We have dwelt on the 
physical pictures and considered what it is reasonable to expect. 
To justify in this manner the final result, it would be necessary to 
make the physical assumptions sharper, in order to draw mathemati 
cal inferences from them. After all that has been done, the final 
result is the equation 
s 
It is, therefore, simpler, both physically and mathematically, now 
that we see what it is reasonable to expect, to begin at this end and 
lay clown equation (7) as the one physical law. 
Unsteady Flow. In the case of an arbitrary flow an instantane 
ous photograph of the lines of flow at one instant would be differ 
ent from that at another instant.* Nevertheless, these lines do not 
shift abruptly, and for a short interval of time succeeding an arbi 
trary instant the rate of flow across S is given approximately by 
(7), or 
s 
where £ approaches 0 with Af. Thus we have 
s 
Again, the reasoning has been heuristic. We have been shooting 
in the target; and now we can bring all the physical assumptions 
considered from the beginning into the one 
Physical Hypothesis. In the case of an arbitrary flow of heat, 
the rate at tohicli heat traverses a fixed surface S is 
s 
* Whereas in the case of a steady flow the lines of flow formed a two-param 
eter family of curves, the lines of flow in the general case correspond to the 
individual particles of the substance at an initial instant, and thus form a three- 
parameter family.