14 Prof. A. Gray: Notes on Hydrodynamics. 



the circuit of which an element; E has length ds, and dl the 

 field-intensity, 



81=7.00.0$, (2) 



and 6T is at right angles to the plane determined by E and P. 

 The resultant intensity at P is the resultant of the complex 

 of vectors SI given by the elements composing the circuit. 



The Amperean rule for the direction of SI is well known : 

 that for the direction of Sq is simpler. Imagine a closed 

 path drawn round the element E, in such a manner that the 

 projection of the path on any plane dues not cross itself, and 

 at E let the path lie along the normal specified. Let a point 

 move round the path in the direction of the rotation at E : 

 the direction of motion at P is the direction of Sq. 



2. Now imagine a circular vortex-filament of infinitesimal 

 cross-section to exist alone in an unlimited incompressible 

 fluid. If a point P be taken in the plane of the filament, the 

 velocity (q) there at right angles to the plane is given by 



d? 

 cos0^, (3) 



-J 



T 



where r is the distance of P from the element E, of length ds, 



and 6 is the complement of the angle 



between ds and the line EP. From 



this we can obtain an approximate 



evaluation of the surface integral of 



flow through a circle coaxial with 



the filament and differing only slightly 



in radius. Let the outer circle, of 



radius a in the diagram, represent 



the filament, and the inner circle, of 



radius a — x, represent the coaxial 



circle, in the same plane. The angle CEB is the angle 6 as 



defined above, and so for the flow, d% say, through an 



element of area rdO dr (EP = r) at P, due to the element E 



of length ds, we obtain 



d X = K ~^-dsd0dr. ..... (4) 



Thus, if we suppose to vary from 3 when EB is along EC, 

 the upper limit is sin -1 {{a — a?) /a}. "We call this 6 X . The 

 limits of r are the two roots of the equation 



i*-2ar cos + a 2 -(a-x) 2 = O. ... (5) 



Approximately, these roots are 2a cos 6, .r/cos 6 ; a closer 



