OF INCOMPRESSIBLE FLUIDS. 445 



In this case p, u and v, are given by the equations 



1 dp du du _ 



- p d X =X - U dx- V dy W' 



1 dp - T dv dv 



- P dj = ¥ - U T X - V dy> < 8 >' 



d u dv n .. 



dl* + Ty = (*>' 



We still have -^ = - , for the differential equation to a line of motion, 



where udy - vdx is still an exact differential, on account of equation (9). 

 Eliminating p by differentiation from (7) and (8), and expressing the 

 result in terms of U, we get the equation which U is to satisfy, viz. 



dU d td*U d 2 U\ dUd^(d*U d*U \ 



dy dx \d& dtf) dx dy (dx 1 + dtp) == ' 



, , . (dU d dU d\ td*U d 2 U\ 

 or, for shortness (^ _ - __)(_+_) = (10). 



In this case, since p = (-£.dx f J^dy), equations (7) and (8) give 



P V- fU dU d * U dUd *U\ j (dU d'U dUd°U\,\ 

 p J\\dy dxdy dx df ) aX+ [dx- dx~dy " ~dy~ tf^J dy ) ' 



tdU d*U^ dU d*U\ 

 \ dx dxdy dy dy' J & ' 

 whence, 



dU d-U dx dU d*U _ ld [(dU\* (dU\*\ 

 dy dxdy dx dxdy * "" s \\ dx I + (dy I ) 



(d*U d*U\ (dU , dU , \ 



-{-dtf + -dtf) Kdx- dx + ~dy- d y) > 



Vol. VII. Paet III. jE 



