Oct. 25. 1SS3] 



NA TURE 



631 



t r this class of motion question 6 must be answered in the 

 affirmative. 



It thus appeared that there was some difference in the cause of 

 instability in the two motions. 



13. Further Study of the Equations of Motion. — Here the 

 author explains that he had so far succeeded in integrating the 

 equations of motion as to find that there must be two critical 

 values of the velocity — the one that at which steady motion 

 would break down into eddying motion, the other that at which, 

 as the velocity diminished, previously existing eddies would die 



oat ; both these velocities depending on the relation U ex. — . 



pc 



14. Results of Experiments on the Law of Resistance in Tnies. 



—The existence of the critical velocity described in the previous 

 article could only be tested by allowing water in a high state of 

 disturbance to enter a tube, and after flowing a sufficient distance 

 for the eddies to die out, if they were going to die out, to test 

 the motion. As it seemed impossible to apply the method of 

 colour bands, the test applied was that of the law of resistance 

 as indicated in questions (1) and (2) in § 8. The result was very 

 happy. Two straight lead pipes, No. 4 and No. 5, each 16 feet 

 long, and having diameters of a quarter and half inch respec- 

 tively, were used. 



The water « as allowed to How through rather more than 10 

 feet before coming to the first gauge-hole, the second gauge-hole 

 being 5 feet further along the pipe. 



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Fig. 8. 



The results were very definite, and are partly shown in Fig. S. 



(I.) At the lower velocities the pressure was proportional to 

 the velocity, and the velocities at which a deviation from this 

 law first occurred were in the exact inverse ratio of the diameters 

 of the pipes. 



(2.) Up to these critical velocities the discharges from the 

 pipes agreed exactly with those given by Poiseuille's formula for 

 capillary tubes. 



(3.) For some little distance after passing the critical velocity 

 no very simple relations appeared to hold between t'.ie pressures 

 and velocities ; but by the time the velocity reached 1-3 (critical 

 velocity) the relation became again simple. The pressure did 

 not vary as the square of the velocity, but as 1 722 power of the 

 velocity; this law held in both tubes, and through velocities 

 ranging fronT 1 to 50, where it showed no signs of breaking 

 down. 



(4.) The most striking result was that not only at the critical 

 velocity but throughout the entire motion the laws of resistance 



exactly corresponded for velocities in the ratio of — . This last 



result was brought out in the most striking manner on reducing 

 the results by the graphic method of logarithmic homologues as 

 described in my paper on thermal transpiration. 



Calling the resistance per unit of length as measured in the 

 weight of cubic units of water i, and the velocity v, log ;' is taken 

 for abscissa, and log v for ordinate, and the curve plotted. 



In this way the experimental results for each tube are repre- 

 sented as a curve ; these curves, which are shown as far as the 

 small scale will admit in Fig. 9, present exactly the same shape, 

 and only differ in position. 



Either of the curves may be brought into exact coincidence with 

 the other by a rectangular shift, and the horizontal shifts are 



Pipe No. a, Lead . 



m - 5. .. • 

 ,, A, Glass . 



000615m. diameter 

 00127 „ 



0*0456 ,, 



Pipe B, Cast Iron ... 

 „ D, 

 ,, C, Varnish 



m. diameter 



°"5 

 o - iq6 



given by the difference of the logarithm of _ for the two tubes, 



the vertical shifts by the difference of the logarithm of — . 



The temperatures at which the experiments had been made 

 were nearly the same, but not quite, so that the effect of the 

 variations of m showed themselves. 



15. Comfarison with Darcy's Experiments. — The definiteness 

 of these results, their agreement with Poiseuille's law, and the 

 new form which they more than indicated for the law of resist- 

 ance above the critical velocity, led me to compare them with 



the well-known experiments of Darcy on pipes ranging from 

 o"oi4 to o"5 metre. Taking no notice of tie empirical laws 

 by which Darcy had endeavoured to represent his results, 

 I had the logarithmic homologues plotted from his pub- 

 lished experiments. If my law was general, then these log. 

 curves, together with mine, should all shift into coincidence 



D 3 

 if each were shifted horizontally through — } , and vertically 



through — 



In calculating these shifts there were some doubtful points. 



