1890.] 



The Iheory of Free Stream Lines. 



131 



It will be observed that they are all of such a character that in 

 going round the boundary of the z diagram we pass but twice from 

 free stream line to rigid boundary. 



It is only in these cases that the actual execution of the work is 

 feasible. 



Case I. — A Jet from a Vessel. 

 The general solution is 



du u—c ' L u—a n 



du) 1 



where —7— = • 



du u—c 



Ex, I. — A rectangular vessel of given width has an aperture in 

 the bottom. 



The elimination of the unknown constants is not possible in 

 general, but if the aperture is symmetrically placed, and d be the 

 breadth of the vessel, c of the aperture, and k of the jet, then 



L ST \k d) d*-k*J 



Ex. II. — Tube projecting into the bottom of a vessel. The 

 simplest results of this case are — 

 (a.) When the tube is very long 



(d- C y = d(d-k), 



with the same notation as in Ex. I. 



(6.) If the tube is of small length Z, and the breadth of the vessel 

 very large 



Case II. — Flow from Pipes. 



An aperture is made in one of the walls of a pipe along which 

 water is flowing. 



The formulae of transformation are 



au 



-l + \/a 3 -l VV-1 



and 



du (u—b) (u—c) 



duo u — a 



du (u—b)(u—c) 



VOL. XLVII. 



