1894.] of equal and opposite hollow straight vortices. 181 



creases from oo to log -y. . The boundary in the Xl-plane is 

 therefore as shown in fig. 3. 



The relation between w and t that transforms the boundary in 

 the plane of w into the real axis in that of t is given by the theory 

 of Schwarz and Christoffel in the form 



tu = kXJdt 



where we have assumed that the points +1, ± k, ±a are those 

 corresponding to F' and I)', G' and C, H' and B' respectively, and 



where a > r > 1. 

 fc 



To integrate this, put ^ = sn w (mod. k), and then 



w = \f du (kW — F sn^ u) 



= \[z{u) + (^-\-¥a'-iyX (1), 



if we chose the coefficient of integration so that w=0 aJd E'. A 

 consequence, utilized below, of this equation is that the points 

 A', I' correspond to the points at infinity on the real axis in the 

 plane of t. Between the three constants, X, k, a, that enter into 

 this equation, we can find two relations. For if ^^ + 1^^^ is the 

 value of w corresponding to G, so that 2<^q is the circulation about 

 the hollow, i/^o the amount of fluid that in unit time flows between 

 it and the axis of x, we have 



<^^ + ,f^ = \\z{K + LK') + {^+¥a'~lYK + iK')Y 



Putting for Z {K ■{■ lK) its value 



j.,fE E' \ 



this is 



<^^ + t^„ = X [[E + (y^V - 1) Z} + i {Fa^K' - E')l 

 giving 



4>, = \[E+{¥a^-l)K] (2) 



t<, = \(A;Vir'-^') (3). 



