192 Dr. Leathern on Two- Dimensional Fields of Flow, 



Similarly a vortex at the point £ = y + z3, round which the 

 circulation is yit, is represented in w by a term 



(Mhr)log{(£-y-i8)/tf—t+'ilt)}. • • (2) 



5. Motion of a ship in a canal. — -As a first example one 

 may consider the case of a doubly pointed ship moving along 

 the central line of a uniform straight canal. On super- 

 position of a motion equal and opposite to that of the ship, 

 the ship becomes part of the fixed boundary. Only half the 

 configuration need be studied, and the z diagram is as shown 

 in fig. 1, wherein the values assigned to f at the important 



points are indicated. The angles at bow and stern are taken 

 to be 2pir and 2qir. 



The velocity at infinite distance being V, there is a source 

 of output IV at ? = a, and an equal sink at f=oo. The 

 geometrical and field relations are 



where E is a positive constant and %? is a curve-factor in f, 

 having linear range from — c to c, and angular range (p + q) it. 

 The curve-factor %> may be any one of several known types, 

 and the shape of the ship depends on the form of ^and the 

 values of ail parameters in the geometrical relation. 



One condition which the parameters must satisfy corre- 

 sponds to the geometrical relation giving the proper breadth 

 to the canal at f=a and £ = » . The abrupt changes in the 

 value of y indicated by this relation lead to 



■E*(«) _,__™ l! .T,„/ »(0 



= / = T ELim-( *S2 I 



(i) 



(a + c)i(a-o) 

 Thus, for example, if the selected curve-factor be 



r=£fS +s H^-*)+(2- c )Xr+e) 1 ~1 !,+5 • • (5) 



it is necessary that 



\\(a^k) + (a-c)Xa + c) l -*\ p+ \a + c)- q (a~-c)-- p =(\ + iy +q . 



