1888.] On the Maximum Discharge through a Pipe. 145 



perfectly independent, and n if they were rigidly and perfectly co- 

 related. The observed value would be almost always somewhere 

 intermediate between these extremes, and would give the information 

 that is wanted. 



To conclude, the prominent characteristics of any two co-related 

 variables, so far at least as I have as yet tested them, are four in 

 number. It is supposed that their respective measures have been 

 first transmuted into others of which the unit is in each case equal to 

 the probable error of a single measure in its own series. Let y = the 

 deviation of the subject, whichever of the two variables maybe taken 

 in that capacity ; and let o^, a? 2 , # 3 , &c., be the corresponding devia- 

 tions of the relative, and let the mean of these be X. Then we find : 

 (1) that y = rX for all values of y ; (2) that r is the same, whichever 

 of the two variables is taken for the subject ; (3) that r is always less 

 than 1 ; (4) that r measures the closeness of co-relation. 



II. " On the Maximum Discharge through a Pipe of Circular 

 Section when the effective Head is due only to the Pipe's 

 Inclination." By HENRY HENNESSY, F.R.S., Professor of 

 Applied Mathematics in the Royal College of Science for 

 Ireland. Received November 15, 1888. 



In the paper on " Hydraulic Problems on the Cross-sections of 

 Pipes and Channels,"* it was shown that the greatest hydraulic mean 

 depth was that for a channel formed by a segment of a circle, and 

 bounded by an arc of 257 .27'. It is easy to find by a similar 

 process the wetted perimeter of a circular pipe corresponding to the 

 maximum discharge when the velocity of the liquid is due only to 

 the inclination of the pipe. 



Among the formula adopted by hydraulic engineers for v, the mean 

 velocity of liquid in a pipe whose hydraulic mean depth is u, we may 

 select Darcy's, which gives 



where a and & are constant coefficients and I a quantity depending on 

 the inclination of the pipe. But as the discharge Q is the product of 

 the mean velocity by the area of cross-section, we have 



* ' Koy. Soc. Froc.,' vol. 44, p. 101. 



