I 



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 rectangular coordinates, x and y ; where moreover no 

 rotating particles are present in 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 {xij) and by v the velocity parallel to y, then, as is well known, two 

 functions of j? and y can be found such that 



_d_^_ dip ] 



'^~ 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 : 



dx'^ dv~~ ?^ dy^ ~ dx^ ?y^~ ^^^1 



For a constant density, /^, and when the potential of the external 

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

 equation — 



'--f+^=^K:)V(|)>i[('|)V(|)'] .... m 



The curves 



//' = constant 

 are the stream lines of the fluid, and the curves 



q) = constant 



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

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

 currents in conductors of uniform conductivity. 



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

 quantity q) + ipi is a function oi x + yi, where i = V^^l. The solutions 

 hitherto found generally express cp and ip as the sums of terms that are 



