124 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
encountered by larger bodies moving with higher velocities through water, or by 
water moving with greater velocities through larger tubes. This discrepancy 
Sir G. Sroxes considered as probably resulting from eddies which rendered the 
actual motion other than that to which the singular solution referred and not as 
disproving the assumption. 
In 1850, after Jouxe’s discovery of the Mechanical Equivalent of Heat, Stoxrs 
showed, by transforming the equations of motion—with arbitrary stresses—so as to 
obtain the equations of (‘‘ Vis-viva”) energy, that this equation contained a definite 
function, which represented the difference between the work done on the fluid by the 
stresses and the rate of increase of the energy, per unit of volume, which function, 
he concluded, must, according to JouLE, represent the Vis-viva converted into heat. 
This conclusion was obtained from the equations irrespective of any particular 
relation between the stresses and the rates of distortion. Sir G. SroKss, however, 
translated the function into an expression in terms of the rates of distortion, which 
expression has since been named by Lord Rayiyrcu the Dissipation- Function. 
2, In 1883 I succeeded in proving, by means of experiments with colour bands— 
the results of which were communicated to the Society*—that when water is caused 
by pressure to flow through a uniform smooth pipe, the motion of the water is direct, 
i.é., parallel to the sides of the pipe, or sinuous, 7.e., crossing and re-crossing the pipe, 
according as U,,, the mean velocity of the water, as measured by dividing Q, the 
discharge, by A, the area of the section of the pipe, is below or above a certain value 
given by 
iKpu/Dp, 
where D is the diameter of the pipe, p the density of the water, and K a numerical 
constant, the value of which according to my experiments and, as I was able to show, 
to all the experiments by PorseuILLE and Darcy, is for pipes of circular section 
between 
1900 and 2000, 
or, in other words, steady direct motion in round tubes is stable or unstable according 
as 
DUT 
Qi tec or > 2000, 
the number K being thus a criterion of the possible maintenance of sinuous or 
eddying motion. 
3. The experiments also showed that K was equally a criterion of the law of the 
resistance to be overcome—which changes from a resistance proportional to the 
* ¢Phil. Trans.,’ 1883, Part III., p. 935. 
