1894.] of equal and opposite hollow straight vortices. 179 



the greater power of the methods of two dimensious enables us to 

 make accurate. 



Here the fluid, which is at rest at infinity, contains two hollows, 

 and its circulations about these are equal and of opposite signs. 

 The hollows themselves move, without change of form or relative 

 position, in a direction parallel to a line which is an axis of 

 symmetry and with a constant velocity F. We shall prove that 

 this motion is possible, and find equations to give the shape of 

 either hollow and the value of V. Impressing on the whole 

 system a velocity equal and opposite to V, and regarding the 

 motion on one side only of the above-mentioned axis of symmetry, 

 we reduce the problem to that of the steady motion of fluid 

 circulating about and flowing past a fixed hollow in a half-plane. 

 Before proceeding with the solution, which is obtained by the 

 method expounded by Mr Love in Proc. Gamh. Phil. Soc. Vol. vii., 

 p. 175 et seq., the cyclic region occupied by the fluid must be 

 converted into an acyclic one. This is done if we draw the 

 shortest distance between the hollow and the straight boundary, 

 which latter we will take as the axis of x, and consider it part of 

 the boundary of the region dealt with. The line just drawn is easily 

 seen to be the other axis of symmetry of the system, which, since 

 the pressure at the surface of the hollow is unaltered by reversing 

 all the velocities, must possess such an axis. We accordingly take 

 it as the axis of y. The boundary in the plane of the complex 

 variable z = x + ty is now as shown in fig. 1, where the arrows 

 show the direction of motion of the fluid. 



ooj- 



B F 



^ B^ c G < zr 



Fig. 1. Plane of z. 



They are inserted on the strength of information given by 

 the case where the hollows are small compared with their distance 

 apart. 



If we define w as ^ + l-^, where ^ and -v/r are respectively the 



velocity-potential and the flow-function of the fluid, it is a function 



of z, as the equations satisfied by <^ and i/r show. Hence if we 



dz 1 



put O = log -T- = log - -I- id, where q cos 6, q sin 9 are the com- 



13—2 



