1888.] Problems on the Cross-sections of Pipes, Sfc. 101 



sharpened Hamiltonian Numbers E n+1 , E», E re _ 1? .... and that con- 

 sequently the relation — 



E w (E ra — 1) E re +i(E n _i — 1)(E M _!— 2) 



may 



be written in the form — 



_ i j_ gC 2 ^- 1 ) _ r(2r-l)(2r-2) s(2s-l)(2s-2)(2s-3) 

 '■ ^ + 2 2.3 + 2.3.4 



_ *(2*-l)(2*--2X2*-3)(2*- 4) 

 2.3.4.5 



w(2w-l)(2w-2)(2w~3)(2tt-4)(2tt-5) 



+ 



2 . 3 .. 4.5.6 



(2.) 



The comparison of this value of p with that given by (1) furnishes 

 us with an equation which, after several reductions have been made 

 in which special attention must be paid to the order of the quantities 

 under consideration, ultimately leads to the determination of the 

 values of A, B, C, . . . . in succession. 



III. "Hydraulic Problems on the Cross-sections of Pipes and 

 Channels." By Henry Hennessy, F.R.S., Professor of 

 Applied Mathematics and Mechanism in the Royal College 

 of Science for Ireland. Received March 14, 1888. 



In that division of hydromechanics which is devoted to the investi- 

 gation of the flow of liquids through pipes and open channels, the 

 resistance due to the friction of the contained liquid against the sides 

 of the pipes or channels has led to expressions for the velocity as a 

 function of the dimensions and shape of the cross-section commonly 

 designated as the hydraulic mean depth. 



This quantity is defined as the quotient of the area of the cross- 

 section of the liquid by that part of its perimeter in contact with the 

 pipe or channel. In a full pipe this perimeter is identical with that 

 of the pipe's cross-section, and in practice this is generally a circle. 



It is also proved from tbe Calculus of Variations that a circle is the 

 closed curve which, under a given perimeter, has the largest area, and 

 by the same processes of analysis a segment of a circle appears to be 

 that which includes the greatest area between its arc and its chord. 



If we call the hydraulic mean depth of a pipe or channel bounded 

 by a curved outline u, its definition gives the condition 



