106 
THE HON. J. W. STEUTT ON THE THEOEY OF EESONANCE. 
giving no normal motion at the surface of the tube. That this is actually the case may 
be easily verified a posteriori , but it is scarcely necessary for our purpose to do so. To 
find the vis viva, 
j u 2 2vrdr= — 9 , 
Jo *y 
5 
u 1 2'zrdrdx— — j dx. 
§ v ' 2 * rdr =%[i (<T 5 )X’ 
Thus 
u 
vis viva = — 
2 7T 
1+^ 
dx. 
The total flow across any section is 7 ry\i=u 0 . 
Therefore 
2 vis viva 1 £ 1 f n , 1 ( dy ^ 2 ~| , 
(rate of flow) 2 ir | ?/ 2 | ~'2\dx) j 
(52) 
This is the quantity which gives an upper limit to the resistance. The first term, 
which corresponds to the component u of the velocity, is the same as that previously 
obtained for the lower limit, as might have been foreseen. The difference between the 
two, which gives the utmost error involved in taking either of them as the true value, is 
(I'll 
In a nearly cylindrical tube ^ is a small quantity, and so the result found by this 
method is closely approximate. It is not necessary that the section of the tube should 
be nearly constant, but only that it should vary slowly. The success of the approxima- 
tion in this and similar cases depends in great measure on the fact that the quantity to 
he estimated is a minimum. Any reasonable approximation to the real motion will give 
a vis viva very near the minimum, according to the principles of the differential calculus. 
Application to straight tube of revolution whose end lies on two infinite planes. 
For the lower limit to the resistance we have 
1 Cdx l l 
ttJ ip ' 4R 1 ^~4R 2 ' / 
ll„ H 2 being the radii at the ends, and for the higher limit 
