156 The N.Z. Journal of Science and Technology. [May 
sinuous the relationship may be expressed by a curve, which tends to 
become asymptotic at a ratio of about 0-82. 
3. The value of id/iv 2 is a function of vd/v for a wide range of values 
of v and of v ; but as regards the diameter the underlying principle demands 
that the roughness should be proportional to the diameter, which is a 
difficult' condition to fulfil. Nevertheless the results obtained by Stanton 
in his experiments on pipes varying from 0-142 in. to 5 in. diameter, but 
having approximately the same surface-characteristic, fall sufficiently 
within limits of experimental errors, although it is not known how far these 
limits may extend if much larger pipes be included in the range of 
experiment. 
4. The function may for analytical purposes be expressed in the form* 
( v \ n „ . . 
— ) + 6 ; and experiments carried out by Landerf on a steel pipe 0’423 m. 
diameter indicate that for the same diameter a change in the surface- 
characteristic making it more or less rough causes a change in both a 
and b and in the index n. 
5. A study of the subject by plotting log. (id/iv 2 — b) against log. vd/v 
would seem to indicate that only one observation is sufficient to determine 
the characteristics of a given surface, whilst Stanton’s experiments are 
suggestive of the idea that observation of the distribution of velocity in 
the pipe will also enable the surface-characteristic to be determined without 
recourse to an extended series of experiments on long pipes. 
6. The possibility suggests itself, upon reading Stanton’s paper upon 
“ The Mechanical Viscosity of Fluids, ”J of extending the usefulness of v 
by taking into account not only the physical viscosity proper, but also 
the quality of the surface and the turbidity of the fluid, and thereby 
including in its scope the. case of water carrying silt and other matter in 
suspension. 
It is evident from a study of the researches referred to that the law of 
the surface friction of fluids, at any rate near enough for practical purposes, 
is expressed for round pipes in the relation between id/iv 2 and vd/v, the 
lack of recognition of which has been in the past the-cause of much con¬ 
fusion and of a vast amount of waste effort in abortive experiments. 
Once the law as stated comes to be recognized by hydraulic engineers 
there will be a tendency on their part to substitute a constant for the 
symbol v, as they are dealing with only one form of fluid, and that at tem¬ 
peratures the range of which usually does not affect the results to any very 
great extent. This tendency will be all the greater, as there is something 
of an abstract nature to most people about viscosity which inclines them 
to ignore it if possible. Such a course would be most regrettable, because 
the symbol serves as a common bond of union for all fluids, gaseous or 
liquid. Moreover, even if one’s interest is limited to hydraulics, there 
remains the possibility, as suggested above, of utilizing and of extending 
the scope of it to embrace a surface effect and the effect of matter in 
.suspension. 
* C. H. Lees, Proc. Roy. Soc., A, vol. 91, p. 46, 1914. 
f C. H. Lander, Proc. Roy. Soc., A, vol. 92, p. 337, 1916. 
x Proc. Roy. Soc, A., vol. 85, p. 369, 1911. 
