Certain Problems of Two-Dimensional Physics. 765 



In the case o£ the ellipse, n = 2, we have the well-known 

 result 



co/£=i(a 2 + b 2 )/a 2 , 



where a-t-6, a— b are the semi-axes. And in the case of the 

 w-cusped hypocycloid, for which b = a/(n— 1), 



»/f=l-2/n, 



As a corollary to the general case we notice that 



ir = ^{x 2 +f-2[a 2 -(n-l)b 2 ]^-2ab[e-^ + (4,/n) sinh ng] cos 7197} 



satisfies the equation V^^^f, which is a particular form 

 of ^ 2 -^r = f(yjr), and is such that the velocity vanishes on 

 the cylinder (2). It therefore represents a possible motion 

 of viscous fluid within this cylinder. 



5. In the case of viscous fluid motions so slow that the 

 squares of the velocities may be neglected, we have 



at p ^x ot p oy 



And in the case of steady motions this leads at once to 



V 4 ^ = 0, (3) 



with the conditions u = v = on the boundary. 



The general solution of (3) subject to these conditions is 

 easily seen to be 



* = "s{ f w- f w-*J i 'x*?} 



together with a similar term, with y, Y in place of x } X, 

 which may be written down by symmetry. 

 In the case of the ellipse, for which 



x i-iy — c cosh (A, + f -f t,rj) , 

 we have 



— sechXcos^cosh (\+f) I F'^) sec rjdrj >+&c. 



In accordance with a result first given by Stokes, it appears 

 to be impossible to determine a solution corresponding to the 



