2 GWYTHER, Velocity -Potential across a Channel. 



problem which has well defined conditions, the irrota- 

 tional motion of a fluid in two dimensions across a channel 

 with straight sides. 



The form of the velocity-potential. 



With the modification of the usual notation adopted 

 in Lamb's Hydrodynamics^ write 



-(l>-i\P =/(x + iy), 

 so that 



u- iv=/'{x + iy). 



The (unction /(x-\- iy) will, in general, contain as arguments 

 the measure and direction, relative to the axes, of gravita- 

 tion, and the coordinates, whatever they may be, of the 

 lines bounding the fluid. As it is assumed that these 

 bounding lines are straight, these arguments may be taken 

 to be ^ and 7,/] and /3i,/2 ^.nd ^^, etc., where/ and j3 are 

 the coordinates of a straight line. 



The origin and axes of coordinates are to be taken 

 arbitrarily, and the principle which is fundamental in the 

 method of this paper is that with any system of boundaries 

 the function /"(;ir-f23/) will enjoy a permanence of form for 

 any change in position or direction of the coordinate axes, 

 or, stated analytically, must be a covariant of the system, 



^ "r iy, t, g, y, pu p2, 5 A, ft , 



or of 



^■^iyyi,g,y,pup^, . A-y, A-y, , 



where y has been selected as the angle by which to 

 orient the system of coordinates in space. 



In considering the conditions for covariancy under a 

 change of axes, the letters x, y, 7,/, j3, will still be retained 

 to mean the quantities relative to the new axes corres- 

 ponding to those they previously indicated relative to the 

 old axes ; after the conditions have been found it will be 

 useful to drop this convention with results which will be 

 seen later. 



