Prof. Helmholtz on Integrals expressing Vortex-motion, 503 



§5. 

 Straight parallel Vortex-filaments. 



We shall first consider the case where only straight vortex- 

 filaments, parallel to the axis of z, exist, whether in an indefinitely 

 extended mass of fluid, or in a similar mass limited by two infi- 

 nite planes perpendicular to the filaments, which comes to the 

 same thing. All the motions are then confined to planes perpendi- 

 cular to the axis of z, and are exactly the same in all such planes. 



We put therefore 



_ du __ dv _ dp __ d V _ 

 ~ dz ~~ dz~~ dz~~ dz~ 

 Then equations (2) become 



(3) become 



a X=o 

 Bt - 



The vortex-filaments thus retain constant angular velocity, so 

 that they also retain the same section. 

 Equations (4) become 



dv ' das 



<m dm 



doc 2 dy 



+ T^=sr- 



By the remark at the end of § 3 we put P=0. The equation of 

 current lines is thus N == constant. 



N is in this case the potential function of indefinitely long 

 lines; and is infinitely great, but its differential coefficients are 

 finite. Let a and b be the coordinates of a vortex-filament 

 whose section is da db, we have 



d'N _ %dadb x—a 



~~ dx it y 2 



. _ dN _ ytadb y-b 

 L ~ dy ~ 7T r 2 



From this it follows that the resultant velocity q is perpendicular 

 to r, which again is perpendicular to the vortex-filament, and that 



tda db 



ff=- 



1 irr 



If we have a number of vortex-filaments whose coordinates are 



2L2 



