100 



boundary conditions are Icnown. Aceordin.iily in wliat follows the velocity- 

 potential will be replaced by the pressure function. 



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

 Avay, 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 = 



P =:; 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: 



. , UT (b — z) 



n = oc smii -c -, „ 



p _ •*BF^ N . sin — i^- 



-2 '-' UTrb ^"^ 



n ^ 1 n^ cosh gn, 



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



. , n-(b — z) 



11 = 00 smli ^ 11 TTx 



4gp_k V ?^. ccs"£^ 



7z n~b ■^^ 



U = 1 11 COSll "9jj^ 



, UT (b 



n = GO cosli 



TV u~b ^a 



u = 1 11 cosh ~2ar 



In the above e(iuations // represents each 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 gpk was taken equal to unity. 

 Making these changes the (Miuations become: 



. , n-(10 — z) 

 n = x smh ^^ ^^^ 



-'^ iiT 2U 



u ^ 1 u- cosh - - 



