November 24, 1892] 



NATURE 



95 



paper show that this formula is incomplete, inasmuch as it does 

 not take any account of the friction of the liquid against the 

 sides of the tube. 



In the first place, if the surface friction is supposed to be 

 zero, so that perfect slipping takes place, the motion is stable 

 for all velocities. If e'^' be the time factor of a disturbance of 

 wave-length A, the value oik is 



2tirVV 

 \ 





.(I). 



where n is a root of the equation Ji(«) = o. 



Experiment shows that when the velocity is greater than 

 about six inches per second, the frictional tangential stress of 

 water in contact with a fixed or moving solid is approximately 

 proportional to the square of the relative velocity. This in- 

 troduces a constant )8, which may be called the coeflicient of 

 sliding friction, whose dimensions are [ML-^J, and are therefore 

 the same as those of a density. This constant may have any 

 positive real value ; /3 = o corresponding to perfect slipping or 

 zero tangential stress, whilst )3 = oo corresponds to no slipping, 

 which requires that the velocity of the liquid should be the 

 same as that of the surface with which it is in contact. Owing 

 to the intractable nature of the general equations of motion of 

 a viscous liquid, I have been unable to obtain a complete 

 solution, except on the hypothesis that ^ is an exceedingly 

 small quantity. This supposition, I fear, does not represent 

 very accurately the actual state of fluids in contact with solid 

 bodies ; but at the same time the solution clearly shows that 

 the instability observed by Prof. Reynolds does not depend 

 upon viscosity alone, but is due to the action of the boundary 

 upon a viscous liquid. 



To a first approximation, the real part of k is proportional to 



A* 



4«- 



(2), 



where 2ir//« is the wave-length of the disturbance, and « is a 

 root of the equation (Jim) = o. Since the second term is a 

 number, this shows that the motion will be stable, provided 



Wafi/fi < a number. 



The experiments of Prof, Reynolds conclusively show that 

 the critical velocity at which instability commences is propor- 

 tional to /i/a ; and the fact that the theoretical condition of 

 stability turns out to be that Wo/^u, multiplied by a quantity of 

 the same dimensions as a density, should be less than a certain 

 number, appears to be in substantial agreement with his experi- 

 mental results. 



The results of the investigation may be summed up as 

 follows : — 



(i. ) TAe tendency to instability increases as the velocity of the 

 liquid, the radius of the tube, and the coefficient of sliding friction 

 increase ; but diminishes as thr- viscosity increases. 



(ii.) The tendency to instability increases as the wave-length 

 (2ir/w) of the disturbance increases. 



The remainder of the paper is occupied with ths discussion 

 of a variety of problems relating to jets and wave-motion. 



I find that when a cylindrical jet is moving through the atmo- 

 sphere, the tendency of the viscosity of the jet is always in the 

 direction of stability. The velocity of the jet does not affect the 

 stability unless the influence of the surrounding air is taken into 

 account ; if, however, this is done, it will be found that it gives 

 rise to a term proportional to the proiuct of the density of the 

 air and the square of the velocity of the jet, whose tendency is 

 to render the motion unstable. The tendency of surface- 

 tension (as has been previously shown by Lord Rayleigh) is in 

 the direction of stability or instability according as the wave- 

 length of the disturbance is less or greater than the circumference 

 of the jet. 



If in addition, the jet is supposed to be electrified, the con- 

 dition of stability contains a term proportional to the square of 

 the charge multiplied by a certain number, «. When the ratio 

 of the circumference of the jet to the wavelength is less than 

 06, n is positive, and the electrical term tends to produce 

 stability ; but when this ratio is gr.ater than o'6, n is negative, 

 and the electrical term tends to produce instability. It must, 

 however, be recollected that when the above ratio is greater than 

 unity the tendency of surface tension is to produce stability ; 



but if the influencing body is capable of inducing a sufiiciently 

 large charge, the electrical term (when 2ira > \) will neutralize 

 the effect of surface tension and viscosity, and the motion w ill 

 be unstable. 



The well-known calming effect of "pouring oil on troubled 

 waters" has passed into a proverb. The mathematical investi- 

 gation of this phenomenon is as follows : — The oil spreads over 

 the water so as to form a very thin film ; we may therefore sup- 

 pose that the thickness / of the oil is so small compared with 

 the wave-length hat powers of / higher than the first may be 

 neglected. Also, since the viscosity of olive oil in C.G.S. units 

 is ahout^ 3-25, whilst that of water is about 0014, the foimer 

 may be treated as a highly viscous liquid, and the latter as a 

 frictionless one. 



The result is as follows : — 



Let pi, p be the densities of the water and oil, T, the surface 

 tension between oil and water, T the surface tension between 

 oil and air, ft the viscosity of the oil, and «*' the lime factor, 

 then, to a first approximation, 



_ _ i^Pi - P) + T,/»-^K.g-p - Tot')/ 

 4m!^Pi-(T-Ti)ot-^} 



For olive oil, Tj = 20-56, T = 36-9, so that T - T, ; and I 

 find that the motion will be stable unless the wave-length of the 

 disturbance lies between about 9/1 1 and 6/5 of a centimetre. 

 This result satisfactorily explains the effect of oil in calming 

 stormy water. 



Oxford. 



University Junior Scientific Club, October 26.— Mr. E. 

 L. CoUis, in the absence of Mr. Bourne, gave an exhibit of 

 Codiuni tomentosum. — Mr. F. C. Britten gave an exhibit of the 

 nest of a trapdoor spider. — Mr. Hill read an interesting paper on 

 the determination of sex, which was followed by a long dis- 

 cussion. — Mr. Fisher exhibited some specimens of crystallized 

 anhydrous oxalic acid, and described their methods of prepara- 

 tion. 



Cambridge. 



Philosophical Society, October 31.— Prof. G. H. Darwin. 

 President, in the chair. — The following officers were elected 

 for the ensuing session : — President : Prof. Hughes. Vice- 

 Presidents : Dr. Cay ley. Prof. G. H. Darwin, Dr. Hill. 

 Treasurer : Mr. R. T. Glazebrook. Secretaries : Dr. Hobson, 

 Mr. J. Larmor, Mr. Bateson. New Members of Council : 

 Prof. Thomson, Mr. F. Darwin, Dr. Shore, Mr. Ruhemann. — 

 The retiring President addressed the Society.— The following 

 communications were made : — Note on the determination of 

 low temperatures by platinum-thermometers, by Mr. E. II. 

 Griffiths and Mr. G. M Clark. The authors, following up the 

 suggestion of Profs. Dewar and Fleming, that the resistance of 

 certain pure metals vanishes at absolute zero, have assumed the 

 possibility of extrapolating the platinum thermometer formulae, 

 and have thus found the temperature at which A' = o. From 

 the previously-published constants of seven different thermo- 

 meters — including Callendar's original wire — the mean value 

 deduced by them is —273^ 86, which is in remarkable agree- 

 ment with Joule and Thomson's thermodynamical value 

 — 273°7. They further suggest that the simple method of 

 determining the resistance in ice and steam and assuming R=-o 

 when /= -273''7 is sufficient to graduate a thermometer con- 

 structed of fairly pure wire, as they show that the errors so 

 introduced will only amount to a fraction of a degree over the 

 range -273° to -f- 150°. — Carnot's principle applied to animal 

 and vegetable life, by Mr. J. Parker. The author discusses 

 the question whether the conditions of the growth of plants are 

 limited by the law of entropy. The assumption is made that 

 Carnot's > rinciple takes account only of the exchange of heat, 

 and the temperature of the material system at which the 

 exchange takes places ; that the differential effect of solar 

 radiation of different kinds consists in variation of the activity 

 but not of the mechanical type of the growth. The increase of 

 available energy due to the building up of inorganic materials 

 into a plant can then only be explained, in conformity with the 

 second law of thetmooynamics, by the aid of differences of 

 temperature during growth : the author gives calculations to 

 prove that the difference between day and night is amply 

 sufficient for this purpose.— Note on the geometrical interpre 

 tation of the quaternion analysis, by Mr. J, Brill. 



« Osborne Reynolds, Phil, Trans. i8£:6, p- '1' 



NO. 1204, VOL. 47] 



