OF INCOMPRESSIBLE FLUIDS. 443 



If we take the value of U, 



U - 2 cos (^ *) F(y) - 2 sin [~ x) f(y), 



and develope each term, such as ay*, in F{y) or y(y), in a series, and 

 then sum the series by the formula 



cos 



n9 + \/ — 1 sin nd = cos"0 (l + r \/ - 1 tan 6 - ....) , 



we find that the general value of U takes the form 



U = lAr" cos (nB + _B). 



As long as the origin of y is arbitrary, only integral powers of y 

 will enter into the development Of F(y) and f(y), and therefore the 

 above series will contain only integral values of n. For particular 

 positions of the origin however, fractional powers may enter. The 

 equation 



d*U 1 dU 1 d*U 



dr 8 r c?r r 1 dff* 



which (5) becomes when transferred to polar co-ordinates, is satisfied 

 by the above value of U, whatever n be, even if it be imaginary, in 

 which case the value of U takes the form 



U =. ^Ar m e n0 cos (mO - log e r" + B). 



We may employ equation (5), to determine whether a proposed system 

 of lines can be a system in which fluid can move, the motion being of 

 the kind for which udx + vdy is an exact differential. 



Let f(x,y) = U\ = C be the equation to the system, C being the 

 parameter. Then, if the motion be possible, some value of U which 

 satisfies (5) must be constant for all values of x and y for which U t is 

 constant. Consequently this value must be a function of U x . Let it 

 = (piUi). Then, substituting this value in (5), and performing the 

 differentiations, we get 



