New Eqtiatiun in Hydrodynamics. 175 



sary that the time should disappear from this equation when- 

 ever for the coordinates ^, y, z are substituted their expres- 

 sions formed with the initial values a, 6, c, and the time /, as 

 Lagrange first noticed. Now as vf/ is supposed to contain an 

 arbitrary function of/, it is clear that this cannot be the case. 

 Besides, if the equation (7.) be exact, as we can augment the 



partial differential coefficient -7- by an arbitrary function of the 



time x(/), it would follow that 



R2 J 11^ dt W 



d^ d^ (,.~ d^ ,..' d'\f 



are at the same time factors which render integrable udx + vdy 

 + ivdz. Hence it ought to be 



# d^ 



d dt d dt a 

 U-J-' -T-. =v J-. , &c., 



or 



u_d^,d^_^^,^ 



V ~ dx dy ~ dxdt dydt* '* 



which would lead to a peculiar form of 4/. 



I subjoin an example which, I think, evidently shows the 

 inconsistency of the equation (6.). Let us suppose the fluid 

 to be homogeneous and incompressible, and of a density p= 1, 

 and let us take 



u=y{t—l), v=x(t+l), «7 = 0. 



The two equations (2.) and (3.) are both satisfied, and the 

 latter gives 



p=f{t) + W-yx-{t^-l)'^+{t^-l)^, 



where /(/) is an arbitrary function of the time, and 



W =/{Xdx + Ydy + Zdz) . 



Now udx-\vdy is not an exact differential, and the factor 



capable of rendering it so is - = — . Then the equation (5.) 



X xy 



becomes 



y(^{t) being an arbitrary function of /; and the equation (6,) 



