161-162] STREAM-FUNCTION. 257 



we find 



cosfl k 2 cos 2 e-l 



{(x - xj + ^ + OT /2 - 2W cos 6}* ~ 2 ( 



where, from comparison of coefficients, 



A = 2/k-k, B = -2/k. 



Hence 



where ^\ (&), ^ (&) are the complete elliptic integrals of the first 

 and second kinds, with respect to the modulus k, defined by (9). 



162. The stream-function for points at a distance from an 

 isolated circular vortex -filament, of strength m', whose coordinates 

 are x', VT', is therefore given by 



The forms of the stream-lines corresponding to equidistant 

 values of ty, at points whose distances from the filament are great 

 in comparison with the dimensions of the cross-section, are shewn 

 on the next page*. 



At points of the infinitely small section the modulus k of the 

 elliptic integrals in the value of ty is nearly equal to unity. In 

 this case we have*)* 



approximately, where k' denotes the complementary modulus 

 (1 k-)%, so that in our case 





* For another elliptic-integral form of (1), and for the most convenient method 

 of tracing the curves ^ = const., see Maxwell, Electricity and Magnetism, Arts. 701, 

 702. 



t See Cayley, Elliptic Functions, Cambridge, 1876, Arts. 72, 77; and Maxwell, 

 Electricity and Magnetism, Arts. 704, 705. 



L. 17 



