100 



boundary conditions are knoAvu. Accordingly in Avliat follows the velocity 

 potential will be replaced ))y the pressure function. 



If the section be horizontal, the problem may be treated in the usual 

 way, but in case the section is vertical the extraneous force, gravity, gives 

 a system of curves which are not orthogonal. 



Let D C = a and A D == b, and suppose the head of water along A B 

 zero. The boundary conditions then to be satisfied are: 



P — when x z== 



P 1= when x = a 



P = h when z ^ b 



w = when z = 



And since the area is a rectangle P, u and w are expressed as Fourier's 

 series : 



. n- (b — z) 

 n =3 cc smh 



P = -^ 2 ^. sin^- 



11 — 1 11^ cosh -;5 



2a 



This differentiated with respect to x and z for u and w gives: 



. - n-(b — z) 



, , n = GO smh „ „ ^ 



4gf)k .^ 2a UTTx 



u = —2^ Z ^ • CCS ~K~- 



n ^ ij-b 2a 



11 — 1 11 cosh ^^ — 



2a 



, n T ( b — z ) 



. , n ^ GO cosh ; 



4gpk ^ 2a . nTTx . 



^^ = -^ S, H^- «"^^^^^'^^ 



11 ^ 1 u cosh 2a 



In the above equations n represents eacli of the successive odd numbers, 

 a and b being the sides of the rectangle may have any desired value. But 

 for simplicity they were in the present case taken equal to ten, and for 

 the same reason g/^k was taken equal to unity. 

 Making these changes the equations become: 



. , n-(10 — z) 



„_ u = 30 smh — 



„ 80 ^ 20 . UTTx 



TT'' , , , IIT 20 



11 =^ 1 u- cosh -- 



