Prof. A. Gray: Notes on Hydrodynamics. 15 



approximation gives 2a cos 6 — x/cos 0, tf/cos 6. If we confine 

 ourselves to the rougher approximation we get easily from 

 the equation 



%= £ 2 f^'f ^, . . (6) 



for the total flow, the result 



X =«a(]ogi a -2). ..... (7) 



The more exact value of the larger root of (.5) might have 

 been used without added difficulty, but a more accurate 

 solution can easily be derived from (7) in another way. 



The order of approximation so far adopted takes that part 

 of the flow through the filament, which escapes passing 

 through the coaxial circle, as equal to that which might be 

 computed by taking the filament as straight. Of course, if 

 the filament is infinitely thin, this part is infinite, but the 

 infinity is avoided in any actual case by taking the cross- 

 section as finite though very small. The flow which passes 

 outside the smaller circle may then be written ica log (#/e) , 

 where e is a very small quantity. The term ica log e also 

 appears in the flow through the circle of radius a, so that 

 ^ for the smaller circle has the value stated in (7). 



Now let the smaller circle be moved out of the plane of 

 the larger through a small distance y while remaining 

 coaxial with the latter. If, then, c be the shortest distance 

 ( = ^/x 2 + y' 2 ) between two points, one on the filament, the 

 other on the circle, the additional flow which escapes passing 

 through the circle of radius a— x is /ca(log c — log x). Hence, 

 to the same order of approximation as before 



x =««(log^-2) (8) 



3. Of course c involves x, but any attempt to calculate from 

 (8) the axial component of velocity at the circumference of 

 the circle of radius a — x would lead to an erroneous result, 

 in consequence of our having neglected quantities of the 

 first order in x. We can now obtain, however, a closer 

 approximation to x by writing, as was done by Maxwell 

 (Electricity and Magnetism, § 705) for the electromagnetic 

 analogue, 



X =*(Alog^ + B), (9) 



and then determining A and B from the physical fact that 



