86 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



Writing this in the form 



du v , du 



m dt =x ~ m -dt' 



we learn that the pressure of the fluid is equivalent to a force 

 m' du/dt in the direction of motion. This vanishes when u is 

 constant. 



The above result may of course be verified by direct calculation. The 

 pressure is given by the formula 



where we have omitted the term due to the extraneous forces (if any) acting on 

 the fluid, the effect of which can be found by the rules of Hydrostatics. The 

 term d<f)/dt here expresses the rate at which < is increasing at a fixed point 

 of space, whereas the value of < in (4) is referred to an origin which is in 

 motion with the velocity u. In consequence of this the value of r for any 

 fixed point is increasing at the rate - u cos 0, and that of 6 at the rate 

 ML Jr . sin 6. Hence we must put 



d(h d\i a 2 . dd> , u sin 6 dti du a' 2 . u% 2 



-j- = -= -- cos 6 - U cos 6 -f- + - -^ = - cos 6 + - cos 2<9. 



dt dt r dr r dO dt r r 2 



