48 Transactions. 



it does not seem to have occurred to any one to combine the two and 

 thereby obtain an approximation to equation (7). 



Kutter's formula for c is too complicated for ready comparison, and, 

 after all, what is required is not a formula for c, but a sufficient number 

 of observations for each class of pipe to enable a curve to be drawn 



correlating — to — The precise form of the equation expressing the 

 v v 



relationship is really only of academic interest. 



Returning to equation (5), the results of the experiments on smooth 

 pipe by Stanton and Panned are plotted in figs. 1, 2, and 3, and indicated 

 by the number 6, whilst the result of Lander's experiments on drawn- 

 steel pipe is indicated by the number 10, the abscissae being values of 



log. and the ordinates values of — . In fig. 4 the same equations 

 v v 2 



/ vs \ vcl 



are plotted in terms of log. (— — b) and log. — . Line 6 represents 



Stanton's experiments, and line 10 Lander's. These two lines converge 

 at or near to a point 0, where the motion changes from linear to sinuous. 

 Line 12 represents linear flow, and should be common to all pipes within 

 limits. The convergence of these three lines indicates that the two pipes 

 fulfil the condition as regards geometric similarity. The method of 

 plotting adopted in fig. 4 affords a ready means of determining the 

 characteristic of any description of surface, provided that the condition 

 before mentioned is fulfilled, for it is only necessary to make one 

 observation of the quantities involved and to join the point representing 

 the observed value to the point in order to determine the whole 

 characteristic. There is one remarkable coincidence between Stanton's 

 and Lander's results — viz., the ratio of a to b is the same in both ; 

 which suggests a possible relationship which would be most useful if it 

 can be proved to have any dynamical significance, but no deduction 

 can be made in the absence of such a proof. 



In applying the principle involved in equation (2) to experiments 

 upon large pipes we encounter several elements of uncertainty. One 

 is that the temperature of the water has not, as a rule, been observed 

 and recorded ; but as the error involved in assuming a uniform tempera- 

 ture and applying it to all the experiments is considerably less than 

 the error arising out of other disturbing factors, and probably less than 

 the error of observation under the conditions prevailing during the 

 experiments, the temperature error is of no great moment. 



Another factor which affects the harmony of the results arises from the 

 fact, that large-diameter pipe lengths are shorter than small-diameter 

 pipes, and that in consequence the joints are more frequent ; and, as the 

 joint is a disturbing element, a large pipe and a small pipe of the same 

 material and surface — such, for instance, as cast iron — are not strictly 

 comparable on account of the increase in the number of joints, and often 

 also because of the different nature of the joint. There is also the 

 possibility that in two experiments on pipes of the same size and material 

 the joint of the one may be better made than that of the other, and 

 greater care taken in aligning the pipes. 



In the case of riveted steel pipes we have still other disturbing factors. 

 The longitudinal joints ma\ be lapped or butted. There may be one, two, 

 or three longitudinal joints in the circumference The circumferential 

 joints may be alternately in and out, or taper ; in neither case is the 

 diameter of the pipe uniform. In one case we have a larger diameter 

 alternating with a smaller diameter by twice the thickness of metal, with 



