132 The Theory of Free Stream Lines. [Jan. 16. 



Let d be the breadth of the pipe, 

 h that of the aperture, 

 I of the jet, 



v 1 the velocity in the pipe before reaching the aperture, 

 v 2 the velocity after the aperture is passed, 

 27 3 the velocity of the jet, 



then d(v l —v 2 ) = lv s , 



% „ _ 1 ^1 + ^2 ! fa + ttp 2 2v 3 — flj — « 2 



If the water is flowing to the aperture equally from both sides, we 

 get 



*=<['4(T'4)'°s£g- 



If one end of the pipe is stopped, we have the case of a jet from 

 the side of a vessel of width d, the aperture being far from the 

 bottom. 



In this case 



, = (±+L\ log *±I +i ± log 2 _^1+ i * Y. 



\Z + d/ *d-l^ 2 d B 2d + l^ 2 \ d*J 



Lastly, if the pipe is very broad, so that we have a broad stream 

 flowing past an aperture, the result is 



* = a. i og ^ +TA /B ; 



Z ^ 3 2 — V ^ 3 V 3 — V X V Vg 2 



« 3 being the velocity of the jet, and v 1 the velocity of the stream. 



Case III. — Impact of a Stream against a Plane. 



The stream impinges at a given angle against an infinite plane. If 

 x be measured along the plane, the equations of the boundaries of 

 the stream are 



x— (1 + a) log cos ^0— (1 — a) log sin a log cos ~) 

 y = v/(l-a 2 ) log cot \(%7r-G) + c J 



and 



x = (l + a) log sin ^0— (1 — a) log cos \0 — a log cos 0") 

 2/ = v/(l-o2) log cot 1(1^-0) + c ' /' 



