OF SMALL DEPTH AND WIDTH. 461 



Substituting these in (4), and still neglecting quantities of the order w\ 

 we set 



^ / . (PJT „ (lA dX 



^dA dX Id^ dy\ dX\ ., 



dx dx [fidx ydx] rfa; K ' 

 equating now separately the coefficients of/' and/", we get 



\dx I gy ' 



drX 



dX Adx fidx ydx' 



dx 



The first, integrated, gives 



X = ± f-^, 

 •' vyg 



and the second 



Hence if we neglect the superfluous constant h\/g, the general inte- 

 gral of (4) is, (•.• ^=/3-*7-i), 



tlierefore, by (c'), 



and the actual velocity of the fluid particles in the direction of the axis 



neglecting quantities which are of the order (<«) compared with those 

 retained. 



3n 2 



