PAPER BY PROF. HELMHOLTZ. 63 



formed, because in an element so easily moved as is the air small dis- 

 turbances can never be entirely avoided. 



It is easy to see that such an endless cylindrical jet, issuing from a 

 tube of corresponding section into a quiet exterior fluid and everywhere 

 containing fluid that is moving with uniform velocity parallel to its axis, 

 corresponds to the requirements of the "steady condition." 



I will here further sketch only the mathematical treatment of a case 

 of the opposite kind, where the current from a wide space flows into a 

 narrow canal, in order thereby also at the same time to give an example 

 of a method by which some problems in the theory of potential func- 

 tions can be solved that hitherto have been attended by difficulties. 



I confine myself to the case where the motion is steady and dependent 

 only upon two rectaugular coordinates, x and y ; where moreover no 

 rotating particles are present iu the frictionless fluid at the beginning, 

 and where none such can be subsequently formed. If we indicate by u 

 the component parallel to x of the velocity of the fluid particle at the 

 point (xy) and by v the velocity parallel to y, then, as is well known, two 

 functions of x and y can be found such that 



__ ijp _ <>£ } 



u ~l)x- Jy I m (l 



= dP = d± j v 



* ~~ dy dx i 



By these equations the conditions are also directly fulfilled that in 

 the interior of the fluid the mass shall remain constant in each element 

 of space, viz : 



For a constant density, h, and when the potential of the external 

 forces is indicated by v, the pressure in the interior is given by the 

 equation — 



^•-*K^®>»[$' + <#] • • • • a.) 



The curves 



ip = constant 



are the stream lines ot the fluid, and the curves 



cp = constant 



are orthogonal to them. The latter are the equi-potential curves when 

 electricity, or the equal temperature curves when heat, flows iu steady 

 currents in conductors of uniform conductivity. 



From the equation (1) it follows as an iutegral_equation that the 

 quantity <p + t/>i is a function of x + yi, where i = y/-l. The solutions 

 hitherto found generally express <p and as the sums of terms that are 



