OF INCOMPRESSIBLE FLUIDS. 449 



The general equations of motion are, 



1 dp y du du du 



p dx dx dy dz 



I dp _ v dv dv dv 



P dy dx dy dz' ^ 



1 dp _ dw dw dw , . 



- -j- = Z — u-j Vj — w-j- ; (13). 



p dz dx dy dz s 



And the equation of continuity is 



du dv dw . 



dx dy dz ' ^ '' 



Putting Wi, ts 2 , tst 3 , for the last three terms in (11), (12), (13), 

 respectively, we have 



p 



— ■" y — f(w\dx + w,,dy + w 3 dz). 

 P 



Hence the pressure consists of two parts, the first, pV, the same as 

 if there were no motion, the second, the part due to the velocity. 

 Now the velocities are given by equation (14), and by the three equa- 

 tions which result on eliminating p from (11), (12), and (13). These 

 latter equations, as well as (14), will be the same as if there were no 

 forces since 



dX = dT dX m dZ and dY dZ. 

 dy dx ' dz dx' dz dy ' 



and therefore we shall not lose generality by omitting the forces in 

 (11), (12) and (13), since we shall only have to add p V to the value 

 of p so determined. 



When the motion is symmetrical about an axis, and in planes 

 passing through that axis, let z be measured along the axis, and r be 

 the perpendicular distance from the axis, and * be the velocity per- 

 pendicular to the axis. Then, transforming the co-ordinates to z and r, 

 and omitting the forces, it will be found that equations (11), (12) and 

 (13) are equivalent to only two separate, equations, which are 



