1891.] Fluid Motions in two dimensions. 199 



and the mean pressure is 



pirV 2 (suc 2 a — sec a) (1 + sin a) 2 



7r (1 — cos a) + 2a sin a 

 where a is determined by (37). 



.(38), 



14. When the sides of the canal are distant, I is small com- 

 pared with d, and if we take Ijd = e we shall have 



so that 



2 « 1 1 3 



a= ^fi 6 -^(^T4) eSnearl y (40) - 



As a first approximation taking a small, we have for the mean 

 pressure 



pirV*($a*) _PttV 2 



a 2 , 4+7T 

 7r 2- +2a 



•(41), 



the same as for a lamina held in an infinite stream. Going to a 

 second approximation we have merely to retain the term 2a in 

 the expansion of (1 + sin a) 2 and this gives for the mean pressure 



PttV 2 



4 + 7T 



4-7T 



4 + 7T 



•(42), 



so that the effect of the sides of the canal is to increase the mean 

 pressure by 



4 P 7r 2 F 2 



(4 + tt) 2 



.(43), 



and the fraction of itself by which the mean pressure is increased 

 is about \e, in which it is to be remembered that e is the ratio, 

 breadth of pier to breadth of canal. 



15. Problem (v). Stream flowing past an obliquely projecting 

 pier. 



Suppose the z boundary consists of parts of two straight lines 

 one of them infinite in one direction and terminated at the point 

 marked 0, and the other finite and inclined to the first at a given 

 angle ir — a, and that the fluid is on the side of the boundary 



