CHAPTER III. 



IEROTATIONAL MOTION. 



31. THE present chapter is devoted mainly to an exposition 

 of some general theorems relating to the kinds of motion already 

 considered in Arts. 18 21; viz. those in which udx + vdy + wdz 

 is an exact differential throughout a finite mass of fluid. It is 

 convenient to begin with the following analysis, due to Stokes*, 

 of the motion of a fluid element in the most general case. 



The component velocities at the point (x, y, z) being u, v, w, the 

 relative velocities at an infinitely near point (x + x, y + y, z + z) are 



If we write 



du 

 -,-, 



dx 



7 dv 



0=-j- 

 dy 



du dw 



dw 



dv 



H/U/M/ \AJVU\ i 



du dw 



1 (dv du\ 



= i [ - + - 1 



2 \dx d) 



dy 







dy 



equations (1) may be written 



u = ax + hy +gz + rfL fy, 

 v = hx + by +fz + Jx - f z, 

 w = tfx + fy H- cz 



(2). 



* &quot;On the Theories of the Internal Friction of Fluids in Motion, 

 Phil. Trans., t. viii. (1845) ; Math, and Pliys. Papers, t. i., p. 80. 



