16 Prof. A. Gray: Notes on Hydrodynamics. 



the flow through the circle of radius a~x due to a vortex 

 filament of strength and coinciding with the larger circle, is 

 equal to the flow through the latter circle due to a vortex of 

 the same strength coinciding with the smaller circle. We 

 assume therefore 



A = a + A x x + , B= — 2a + B 1 x+..... . (10) 



Since the vortex filament fas it was in the case considered 

 by Lord Kelvin) is to be taken of small though finite cross- 

 section, we need not carry the calculation beyond terms 

 involving the first power of the ratio xla. No first power 

 of y, or indeed any odd power, can enter, since the value of % 

 cannot be altered by changing the sign of y. 



We substitute then in (9), with the values of A and B 

 from (10), a — x for a and — x for x, and obtain 



K 1 1 x\, Sa 1 x } ,.,.,. 



4. We now consider a vortex rino- of small circular cross- 

 section (radius r) made up of thin coaxial vortex filaments, 

 each of the same strength per unit area of section, and 

 calculate the flow through the circle, of radius a, which 

 forms the circular axis of the anchor ring surface of the 

 assemblage of filaments. Thus for the different filaments 

 c varies from to r. 



To find this axial flow, we calculate the rate of change of 

 ^ when the ciicle through which the flow is taken is 

 widened, while the filament producing it remains fixed. 

 That is, we have to differentiate ^ with respect to x while 

 H = a — x remains unchanged. Thus, substituting R-f^ for a 

 in (11) we get 



x =«{(R + i*)log®^)-2B-§*} . (12) 

 and therefore 



-!«-')— (i + ?) + i-?- 



The axial component Sv of velocity at any point of the circle 

 of radius a is this divided by lira. Thus 



5 k /' 8a \ k (I 2a\ k x" , 10 , 



