VIII. 



THE THEORY OF FREE LIQUID JETS.' 



I'.v Prof. G. KlRCHHOFF. 



Helmholtz in his communication on discontinuous motions in liquids, 

 Berlin, Monats-berichte, April, 1868,t has for the first time determined 

 the form of a free jet of liquid in a special case. The method used by 

 him in this determination can, as will here be shown, be so generalized 

 that it leads to the solution of the same problem for a large number of 

 cases. 



It is assumed that the fluid is incompressible, that no exterior forces 

 act upon it, that its particles do not rotate, that the currents are steady, 

 and finally, that the movement is everywhere parallel to a fixed plane. 



Let x and y be the rectaugular coordinates of any point of the space 

 occupied by the flowing liquid reckoned parallel to the fixed plane and 

 let cp be the velocity potential at this point, then cp is a function of x 

 and y such that it satisfies the equation 



In this equation C ^L ant i iCf) are the velocities parallel to the axes of 



jx jy 



x and y and if p is the pressure and /? is the density, then we have 

 further 



>=°-my<m 



where c indicates a constant. If the flowing liquid has a free boundary 

 then this must correspond to a stream line and the pressure must be 

 constant throughout it. The second of these conditions, if we adopt a 

 proper system of units, will be expressed by the equation 



f ) +(£)"-» 



* From Borchardt's Journal, 1869, vol. lxx, or Kirchhoff Gesammelte Abhandlungen 

 Leipsic, 1882, pp. 41G-127. 



t [See also No. Ill of this present collection of Translatious.l 

 130 



