1894.] of equal and opposite hollow straight vortices. 187 



choosing the constant of integration so that the origin is at the 

 point H which corresponds to ^= 1. 



The free surface is given by negative values of t. Putting 

 therefore t=—u, and dividing the equation into two by consider- 

 ing separately the real and imaginary parts, 



x = — -jj (log u — u — 5), 



where the value of the positive quantity — yr is determined by the 



breadth of the jet. Since the value of y corresponding to E 

 is infinite, we conclude that the hollows of our original problem 

 are, in the case when they are very close, much flatter on their 

 near than on their remote sides. 



axis of symmetry 

 Fig. 7. 



In fig. 7, is drawn an intermediate case of a pair of vortices in 

 which k has the value sin 89° = '9998. The Elliptic Functions 

 are taken from Legendre's Tables. Only one hollow is drawn, 

 the other being the image of this in the axis of symmetry. It is 

 remarkable that, even for a value of k so near the limit unity as 

 this is, the limiting case of linear hollows is not closely approxi- 

 mated to, although the velocity of translation, which is here 

 = -97 U, differs little from its limit U. 



The artifice of making the region occupied by the fluid acyclic 

 may be applied to any other case of the circulation of fluid about a 

 hollow, provided that the system has an axis of symmetry. The 

 only other case, besides that here treated, in which the integrations 

 can be effected, seems to be that of a stationary hollow between 

 two parallel planes, a case discussed by Prof. Michell in Phil. 

 Trans. 1890. 



