4 GWYTHER, Velocity-Potential across a Channel. 



which can easily be seen to satisfy the requirements which 

 have here been developed. 



In the case of a channel bounded by 2, 3, etc., sides 

 we have found what arguments may enter into /(;ir + 2J/), but, 

 of course, from general considerations we cannot learn 

 more of the manner in which they enter the function. We 

 proceed to deal with other matters with regard to which 

 general conclusions can be arrived at. 



Simplification of results consequent on a 

 change of axes. 



In this process a very obvious advantage resulting 

 from the stipulation that f{x-\-iy') shall have its general 

 covariant form becomes apparent. 



The convention which has been adhered to up to the 



present that 7, j3i, ^^^ , (as well as x-\-iy) shall, after a 



change of axes, represent magnitudes measured from the 

 new axes, will now be dropped, and in the case of the angles 

 (but not oi x-^-iy) will be considered to retain the reference 

 to the original axes. This will, of course, make no change 

 in j3 — 7, but in 7 only. 



The change necessary, when the axes are turned 

 through an angle a, will merely be to write 7 — a in place 

 of 7, or sometimes more simply, to write (;r+2/)^«« in place 

 o{{x-\-iy). 



Condition at a boundary line. 



Supposing now that a convenient system of axes has 

 been chosen in any case, and that the angles are given 

 relative to that system, it becomes necessary to express 

 the condition that there is no flow over the boundary 

 whose coordinates are /, |3, or that ;// is constant along 

 that boundary. Changing to an axis of y parallel to this 

 bounding line, from the last paragraph it appears that 

 all that is necessary to ensure this condition, is to write 



