1891.] 



Fluid Motions in two dimensions. 



183 



to the further sides of the vessel, and finally the underside of 

 the line 9 = — \ir from oo to the point corresponding to u = c. 

 Hence the boundary is 



1 6i=0 



ft 00 



-1 



where the point marked 1 is the origin in the fl plane and all 

 points are marked by the corresponding values of u, and the 

 figure is traced continuously by starting from any point, say c, 

 going from c to a l} from a 1 to c x , from c x to a 2 , and so on, keeping 

 the region marked out on the left. 



5. The H region can be conformably represented on the half- 

 plane u by means of the relation 



u —c (u — c,) (u — O ... 



(2), 



dn = A 



du VC" 2- 1) (u — «j) (u — a 2 ) ... 



where the number of the letters a v a 2 , ... exceeds that of the 

 letters c v c 2 , ..'. by two, and A is a constant which can be de- 

 termined ; for the angles at 1 and — 1 in the H plane are each 

 \tt, the angles at a v a 2 , ... are each 0, and the angles at c, c ,... 

 are each 2tt. 



Now the function 



(u-c)(u-c,)(u-c 2 )... 

 can be put into partial fractions in the form 



u — a. 



+ 



A, 



u — a„ 



+ 



where A lt A 2 ,... depend only on the c's and «'s, and thus we 

 may write (2) in the form 



du 



= -AiX 



L(t»- O|i ).v(i-t0 



■(3), 



and the integral of this is 



n = -Ailog\B\J 



u — a 



where B is a constant which can be determined. 



A nlsJi l - "nh 



J 

 (4), 



