OF THE MOTIONS OF FLUIDS. 295 



vanation if -r- 6x+^t-cv-\ — r—oz is an exact variatioD 

 at at ^ at 



in the first instance : now we Lave from the equation (H) in 

 this case ^Jx + ^%+-^ S. = 8F- xg J (-^>+ 



( -T— ) + ( -p- ) ? -' the first member of the equation is 



consequently an exact variation of a function of or, y, and 2; 



the function udx + vdy-\-w^z is therefore an exact variation 

 in the subsequent instant if it is in the preceding : it is 

 therefore an exact variation at every instant. 



370. Theorem. When the motion of the 

 fluid is infinitely small, we have V —f-^ =:-£- 



Neglecting the squares and products of u, v, and w, 

 the partial velocities, the latter part of the equation (H) 

 (366, 368) will vanish; and in this case udx + v^y + w^z::^ 

 ^(p must be an exact variation whenever jp is a function of 



f : and when the fluid is homogeneous, the equation of con- 



. . . ^ dd^ dd^ . dd^ ^, 



trnuity remains 0=z -^ + — ^^—. These two 



equations contain the whole theory of infinitely small un- 

 dulations of homogeneous fluids. 



§ 34. Case of the rotation of a homogeneous fluid 

 mass, with a uniform velocity, round one of the axes of 

 the coordinates. P. 97. 



371. Theorem. In the case of a homo- 

 geneous fluid, revolving round an axis with a 

 uniform velocity, the equation of the pressure 



