1894.J of equal and opposite hollow straight vortices. 183 



From (1) -r- =k\(a^ - sn^ u), and so 

 ctu 



= ^ [(2^=^ - 1) dn' ti - {2a^ - a'Jc' - 1) - 2ta Va' - 1 F sn w en w 



+ 2ia'^a^ — 1 dn ti 



+ const. 



Now z has the same value for D and F, that is, for m = iT and 

 u = — K. This requires that the coefficient of ii should vanish, 

 that is 



(2a^ - 1) (^ - l) + ^'^' = ^' 



or 



2_ K — E . . 



^ ~'2(K-E)-k'K ^^' 



Further, since z = if u = K + lK', the value corresponding to 0, 

 we find that the constant of integration is 



Putting this value in the formula for z, and equating separately 

 the real and imaginary quantities involved in the resulting 

 equation, 



a;=^{2a'-l)Z{u) (6), 



y = ^i(2a' -1) ~ + 2a^/^r^ dnul (7). 



These equations with those numbered (1), (2), (3), (4) and (5), 

 give the form of the free surface and the other details of a complete 

 solution of the problem we started with. 



We can in two special cases find the form of this curve without 

 the aid of tables of elliptic functions. The first is that of k small. 

 Using the approximate formulae 



K = l(l + ^^^+^k' 

 2 V 4 64 



^ = I(l-f-6V)' 



