r 



446 Sir William Thomson on Stationary 



free surface of the water, and four vertical planes, two of 

 them, called A , A, perpendicular to the stream, and two of 

 them parallel to the stream and at unit distance from one 

 another. Let ty P B, and ty B be vertical lines on the two 

 transverse ends A and A of the space S ; ^}, $ being points 

 of the surface, and B, B points of the bottom. Let 



#B = D and $P=y, 



and let u be the horizontal component velocity at P. The 

 rate of delivery of momentum (per unit of time understood) 

 from S by water flowing across A is equal to 



'D 



u?dy (i); 



o 



and the excess of delivery of momentum from S across A 

 above receipt of momentum across A is equal to 



§\*dy- { j ° v?dy | (2). 



When this is positive, the water between A and A must 

 experience, on the whole, a pressure in the direction from A 

 towards A, made up of difference of fluid-pressures on the 

 end sections A and A, and pressures upon the water by 

 fixed inequalities, if there are any, between A and A. 

 Hence if X, X denote the integral fluid -pressures on the 

 ideal planes A, A , and F the sum of horizontal pressures of 

 the inequalities on the fluid, regarded as positive when the 

 direction of the total is from A towards A , (2) must be equal 

 to 



X -X-F (3). 



Hence we have 





F = 



|x+f D w %j-(x+rv^ . . . (4). 



Now the fluid-pressure at P is equal to #y + i(C) 2 — <f), by 

 the elementary formula for pressure in steady motion (the 

 pressure at the free surface being taken as zero), q and q 

 denoting the velocity of the fluid at ty and P respectively. 

 Hence 



S=J o D [OT + l(q 2 -5 2 )]rfy=^D + q 2 )D-^y-% . (5). 



Hence 



X + fVtfy=KffD+ 2 )D+I fV-^y • • . (6), 

 Jo Jo 



if v be the vertical component velocity at P. 



