506 Prof. Helmholtz on Integrals expressing Vortex-motion. 

 and sia e, and adding and subtracting, 



L sin e — M cos e= — — I I I — - -gdg d(e—e) dc, 



L cos e + M sin e= + ^— I I I -gdg d(e—e) dc. 



In both integrals e and e appear only in the form (e—e) ; and 

 this may therefore be taken as the variable in the integration. 

 In the second integral the element for e— e = Z is destroyed by 

 that for e— e = 2ir— t ; it is therefore zero. If we put 



4r= — fff cr cose gdg dedc . 



r 27rJJJ v / (^-c) 2 + % 2 ^-^-2^%cose , * ; 

 we have 



M cos e — L sin e = ijr, 



Msin e + Lcose=0; 

 or 



L= — yfrsine, M = i|rcose (7 a) 



Calling t the velocity in the direction of the radius %, and no- 

 ticing that on account of the symmetry about the axis of z there 

 can be no velocity in the direction of the circumference, we have 



w = TCOse, v = Tsine; 



and from equations (4) 



_ _dM _dh _^M_^. 



dz dz ~~ dx dij 



From this 



dz d X X 



or 



vc- d -W- •*-*$?■ ■ ■ ■ ■ ™ 



The equation of the current-lines is therefore 



^% = constant. 



If we perform approximately the integration for -\\r for a 

 vortex-filament of indefinitely small section, putting adg dc=m l9 

 and the corresponding part of yfr = yfr mii we have 



where 



is+xT + iz-c? 



