OF INCOMPRESSIBLE FLUIDS. 441 



When udx + vdy -rivdz is an exact differential, equation (1) may be 

 deduced in another way*, from which it appears that C is constant. 

 Consequently, in any case, udx + vdy + wdz is, or is not, an exact 

 differential, according as C is, or is not, constant. 



Steady Motion in Two Dimensions. 



I shall first consider the more simple case, where udx + vdy is an 

 exact differential. In this case u and v are given by the equations 



Now from equation (3) it follows that udy — vdx is always the exact 

 differential of a function of x and y. Putting then 



dU = udy — vdx, 

 U = C will be the equation to the system of lines of motion, C being 

 the parameter. U may have any value which allows -j— and - -3— to 



satisfy the equations which u and v satisfy. The first equation has 



been already introduced; the second leads to the equation which U 

 is to satisfy; viz. 



d*U d*U n , K , 



dx^ + W = ° -^ 



* See Poisson, Traite de Mecanique. 



