334 Prof. L. Vessot King on 



when the rapping ceased ; the phenomenon of reformation 

 of stream-lines at higher velocities was observed both by 

 the colour-band method and by a special thermal method. 

 It was also noted, in passing, that the amount of colour 

 employed had a marked effect on the velocity at which the 

 colour-band broke up. According to the views expressed 

 in the preceding section, the reformation of stream-lines at 

 the higher velocities would correspond to the excitation of 

 symmetrical transverse modes of higher frequency than the 

 indeterminate mode giving rise to the first appearance of 

 turbulent flow 31 . 



Among the more accurate observations carried out for the 

 distribution of velocity over the cross-section of a circular 

 pipe through which water was flowing, may be mentioned 

 the experiments of Morrow 32 . Measurements were carried 

 ■out by means of a Pitot tube, and it was found that the 

 distribution-curves changed gradually from approximate 

 paraboloidal distributions to those typicul of velocities above 

 the " critical velocity.'''' 



The scope of experiments of this type was extended by 

 Stanton 33 , and recently in an exhaustive memoir by St-nnton 

 and Pannell 34 . The distribution of velocities over the cross- 

 sections of a number of circular pipes was accurately 

 determined for air- and water-flow above the "critical 

 velocity" in order to obtain evidence over as wide a range 

 -of density and viscosity as possible as to the accuracy of 

 Payleigh's Principle of Dynamical Similarity to cases 

 involving surface-friction between solid and liquid surfaces. 

 IE R be the resistance per unit area, v the kinematic viscositv 

 of the fluid (y = fju/p), and p its density, Payleigh's Principle 35 

 gives in its application to the particular case considered, 



R = pW¥(UD/v), ..... (21) 

 where D is the diameter of the pipe, U the mean velocity 



31 A close examination of Reynolds's curves (Scientific Papers, vol. ii. 

 diagram 1, p. 90) connecting velocity and pressure-gradient indicates 

 the existence of discontinuities at higher velocities than that taken as 

 the critical velocity. 



32 Morrow, Rov. Soc. Proc. vol. lxxvi. a., p. 205 (1905). 



33 Stanton, T. E., Roy. Soc. Proc. vol. lxxxv. a, p. 366 (1911). 



34 Stanton & Pannell, (l Similarity of Motion in Relation to the Surface 

 Friction of Fluids," Roy. Soc. Trans, vol. ccxiv. a, pp. 199-224 (1914). 

 See also a discussion of these results by Lees (Roy. Soc. Proc. vol. xci. a, 

 p. 46, Nov. 1914), who derives from these observations accurate 

 •empirical formulae based on the Principle of Dyuamical Similarity, 

 exDressing the relations between mean velocity and pressure-gradients 

 for the flow of water and air in pipes of circular cross-section. 



35 Rayleigb, see footnote (19). 



