OF THE MOTION OF FLUIDS. ' 175 



First, it may be observed that u and v are not both possible for any 

 values of x and y, unless the functions F' andy be the same. Again, 

 as the form of F' we are seeking for is to be independent of all that 

 is arbitrary, it will remain the same whatever direction we arbitrarily 

 assign to the axes of co-ordinates. Let therefore the axis of y pass 

 through the point to which the velocities u, v, belong. Then 



y = 0, u = 2F'{x), v = 0. 



If now the axes be supposed to take any other position, the origin 



remaining the same, u will be equal to / ^ ^ F' {^/x^ + y^). 



Hence 



F'{x + y^^) + F'(x-yV-l)=-^^y^^=^.F'(./^FTr), 



a functional equation for determining the form of F'. Let 



x + yy^ - l=m, and x — y^/— 1 =n; 

 then 



2x = m + n, and "s/ x^ -k-y^^s/ mn. 

 Therefore, 



c 



It is easily seen that if F\y/mn) =— f=, the equation is satisfied. 

 Hence ^ 



^ = ^-7=+ ^== = A^ and^--^^. ^-^ 



dx x + yV^l^ a;-yV-l x'^ + y^' dy~x^ + y'-' 



2C -^ - 



and consequently the velocity at xy, or \/ u'^ + v^ = —-r=^ ^- ,., ,, 



These results shew that the velocity is directed to or from the origin 

 of co-ordinates, and varies inversely as the distance from it. But we 

 must observe that this limitation as to the point to which the velocity 

 is directed, is owing to the particular forms, x+y'\/~^, x-ys/ — \, 



z2 



