PAPER BY PROF. HELMHOLTZ. 53 



Therefore if we put 



1 f f r (7 cos e (7 d(i de dc 



"^ ^ ^ J J J '^^W^WWW^W^^e ^"^^ 



then will 



.1/ cos £ — L siu e = if: 

 M sill e -\- L cos e = 0, 



or L = — ij' siu £, M = if; cos £. (7a) 



Let r deuote the velocity iu the direction of the radius Xi ^ii<^ t;ou- 

 sider the fact that on account of the symmetrical position of the vortex 

 rhig" in reference to the axis z the velocity must be zero in the direction 

 of the circumference of the circle, we must have 



uz=T COS f, v = r sin e 



and according to equations (4) 



Hence it follows that 



or 



(76) 



Therefore the equation of the stream line is 



tf: J = const. 



When we execute the integrations indicated in the value of i/-, first 

 for a vortex filament of infinitely small cross-section, putting therein 

 m^ — G dg dc and indicating by /p^ the part of t/- depending thereon, we 

 have 



wherein F and E indicate the complete elliptic integrals of the first 

 and second order respectively for the modulus h. 

 For brevity we in\t 



U=^ {F-E)-hF, 



H 



where U is therefore a function of ;c, then is 



JU z—c 



T^X 



=™V* 



If now a second vortex filament m exist at the point determined by 

 X and z, and it we let ri be the velocity in the direction of g that ni 

 communicates to the filament Wi. we then obtain the value of this ve- 



