212 MR. S. S. HOUCIH ON THE APPLICATION OF HARMONIC 
found by integrating the above expression with respect to y between the limits y=y\, 
y = To- 
Thus, the rate at which liquid is entering the elementary volume PPt' across the 
face PQ' is expressed by 
Sy8 dy. 
* Ti 
If in this expression we change a into a + 8a, we shall obtain the rate at which 
fluid is flowing in the positive direction across the face SP'; therefore, the rate at 
Avhich fluid is leaving the element across the face SP' is expressed by 
8^ fdU/Vis) dy + Sa 8^ F Ildu/A^A,) c«y| • 
In like manner, the rate at which fluid enters across the face PS' is 
8a r(V/;. 3 /.i) dy, 
Yi 
and that at which it escapes across the face QP' is 
8« nY/hJ,,) dy + SaS/3^ {fV/Vh) • 
Lastly, in virtue of the boundary-equation (4), which holds at the surface 
P'Q'PdS', the rate at which fluid enters over this surface is zero, while in virtue 
of (5), which holds at the surface PQPS, the rate at which fluid escapes over the 
latter surface is expressed by 
Su 8 ^ _ 
/q h .2 dt 
Xow the total amount of liquid contained within the elementary volume under 
consideration is constant, and therefore, if we equate to zero the sum of the rates at 
which fluid is entering over all the six faces, we obtain the equation of continuity in 
the form 
or 
{fyUAA)f*r} — a} - q ;f af - 
I • • (10)- 
So far no approximation has been made other than that involved in supposing the 
