Prof. Helinholtz on Integrals expressing Vortex-motion. 505 



riite distance, they travel forwards with equal velocity, and in 

 parallel directions perpendicular to the line joining- them. 



To this last case may also be referred that in which a vortex- 

 filament of indefinitely small section moves near an infinite plane 

 to which it is parallel. The condition at the limits (viz. that 

 the fluid must move parallel to the plane) will be fulfilled if in- 

 stead of the plane there be an infinite mass of fluid with another 

 vortex-filament the image (with respect to the plane) of the first. 

 From this it follows that the vortex-filament moves parallel to the 

 plane in the direction in which the elements of the fluid between 

 it and the plane move, and with one-fourth of the velocity which 

 the elements at the foot of a perpendicular from the filament on 

 the plane have. 



With rectilinear vortex-filaments the assumption of an indefi- 

 nitely small section leads to no inadmissible consequences, since 

 each filament exerts upon itself no displacing action, and is only 

 displaced by the action of other filaments which may be present. 

 It is different with curved filaments. 



§ 6. Circular Vortex-filaments, 



Let there exist in an infinite mass of fluid only circular vortex- 

 filaments whose planes are parallel to that of xy, and whose centres 

 are in the axis of z, so that all is symmetrical about that axis. 

 Let us change the coordinates by assuming 



#=^cose, a=g cose, 



^ = %sine, b=g sine, 



z=z, c—c. 



The angular velocity a is, by the above assumption, a function 

 of x an d z or of g and c only, and the axis of rotation is per- 

 pendicular to ^ (or g) and axis of z. The rectangular compo- 

 nents of the angular velocity at the point g, e, c are, therefore, 



f=— or sine, 77 = 0- cose, f=0. 



In equations (5 a) we have 



r* = (z-c)* + X *+g*-2 X $cos (e-e), 



N = 0. 



From the equations for L and M we obtain, multiplying by cos e 



