MOTION IN RELATION TO THE SURFACE FRICTION OF FLUIDS. 
209 
inlet to the pipe to ensure steadier conditions, and making a series of observations of 
surface friction, the velocity at which the flow became fully eddying, as shown by the 
ordinates of the curve reaching the same value as for the other pipes, was almost 
precisely that found by Reynolds for his upper limit, i.e., at a value of vd/v — 16000. 
When the mouth-piece was removed and the fluid, which in this case was air, was 
drawn into the open end of the pipe from the room, by the suction of a gas holder, 
the observations plotted in fig. 3 were obtained. These show a critical value of 
vdfp of about 3000. Beyond this value the results are in complete agreement with 
the other pipes. Further work in this direction is being made the subject of a 
separate investigation. 
In the experiments on the thick oils it was found impossible, with the pumping 
appliances available, to reach the critical value of the velocity, so that the experi¬ 
ments were all confined to cases of stream-line motion. The particular oil on which 
these observations were made had a value of the kinematical viscosity at 15'5° C. of 
36'2, or 3230 times that of water at the same temperature. By heating this to 50° C. 
the kinematical viscosity could be reduced to 2'1, but even at this temperature the 
critical velocity in the 10 cm. pipe used for the experiments would have been 525 cm. 
per second. As a matter of interest the results of a series of observations of the 
surface friction of this oil, when flowing through a steel pipe 10T cm. diameter at 
speeds varying from 5 to 60 cm. per second, are given in Table IV. and are also plotted 
in fig. 3. As the value of R fpv 2 in these observations is in some cases as much as 400 
times that of its maximum value in the air and water experiments, a separate diagram 
has been made in the right-hand corner of fig. 3. In this diagram the curve is drawn 
from the equation —„ — = 8. A table of viscosities determined by a viscosimeter 
pv 2 v J 
has been used in the calculation of the values of vd/v for the abscissae of the plotted 
points. Since, in all probability, the apparatus used for the experiments constituted 
a more refined method of determining the viscosity than the viscosimeter itself, any 
systematic variation of the plotted points from the curve would not necessarily 
indicate errors of observation, but it will be seen that the agreement is remarkably 
good. 
The Limits of the Index Law of Resistance. 
The determination, in these experiments, of the frictional resistance of a 1'255 cm. 
pipe, when the velocity has ranged from the first commencement of eddying motion, 
at a speed of flow of 22 cm. per second, up to a speed of 3150 cm. per second, has 
made possible a check, within these limits, of the accuracy of the well-known index 
law. This law appears to be due mainly to the observations of William Froude and 
Osborne Reynolds, and there is no doubt that over a moderate range of speed it 
holds with considerable accuracy. It may be mentioned that the ranges of speeds 
obtained by Osborne Reynolds in his experiments on 0'62 and 1'27 cm. pipes were 
VOL. CCXIY.- A. 2 E 
