PAPER BY PROF. HELMHOLTZ. 53 



Therefore if we put 



1 f f f G cos e 9 d& d e d° 



ZxJJJ V{z-c)*+x'+(f-2gXcose W 



tben will 



M cos e — L sin e = ip 

 M sin a + L cos 6 = 0, 



or L = — ?/,' sin s, M = ip cos £. (7a) 



Let r denote the velocity in the direction of the radius j, and con- 

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

 ring in reference to the axis z the velocity must be zero in the direction 

 of the circumference of the circle, we must have 



u=r cos f, v=r sin e 



and according to equations (4) 



Hence it follows that 



dip jtp i// 



dz* ?X^ x 



or 



r* = -if> « X J-<§£. <7») 



0" O A. 



Therefore the equation of the stream line is 



ip X — const. 



When we execute the integrations indicated in the value pf y, first 

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

 m x = a dg dc and indicating by ip m the part of ip depending thereon, we 

 have 



±9X 



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

 and second order respectively for the modulus x. 

 For brevity we put 



U= 2 (F-E)-hF, 



It 



where U is therefore a function of «, then is 



Ml lay ?U z ~ c 



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

 X and 2, and if we let t x be the velocity in the direction of g that m 

 communicates to the filament m u we then obtain the value of this ve- 



