Certain Problems of Two-Dimensional Physics. 763 

 So that 

 yjr = \ {F(<9) - F(t) } + i&U(sin + sin t) - iaV(cos + cos r) 



alt 



— Jwc 2 (2 cosh 2 A, + cos 26 -f cos 2r) , 



where a = c cosh A, 6 = c sinh X. 



Hence, finally, omitting a constant, 



yjr = e~^(bV sin tj — aV cos rj) — ^coc 2 e~ 2 % cos £77, 



4. In the case of a vortex filament bounded by the curve 

 (1), we have, within the vortex V 2 ^t=2?, and without the 

 vortex \7 2 ^jr o =0, where £, supposed constant, is the vorticity. 

 And on the boundary we have 



where it is assumed that the vortex rotates with constant 

 angular velocity to. At infinity yfr must take the same form 

 as the gravitational potential of a cylinder of the same form 

 and of density —2ir^. Further, in order to avoid infinite 



velocities, -~~ and -^ must vanish at the singular points 



of the transformation 



z = x + ty = X(r) + lY(t), 



namely, the points where j- = 0, i. e. 



X'(t) = -*Y'(t), 

 and therefore X'(0)=iY'(0). 



We easily find 

 +t-l«*+j)- ? -j!(X 1 ' + X, ! +Y 1 ' + Y/)+ ^{F(r)-F(tf)}, 



^=i»(X 1 » + X,' + Y 1 ' + Y,»)+ \ t f (XY'-X'Y>fc, + J- {F(t) -F(ff)}, 



where F is to be determined by the conditions given above. 



The case of variable vorticity may be treated in the same 

 way. It is evident, however, that the problem is precisely 

 the same as that of determining figures of equilibrium of 

 rotating fluid, where co is put for — to 2 and f for — 2irp. 

 (See §§ 10 and 11 infra.) 



3 D2 



