CHAPTER III. 



IKROTATIONAL 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 (#, y, z) being u, v } w, the 

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



(1). 



If we write 



_ du , _ dv _dw 



~ dx' ~ dy' ~ dz' 



dw dv\ __ l fdu dw\ , _ ^ fdv du 



dw dv\ , fdu dw\ , fdv du 



equations (1) may be written 



u = ax + hy + gz + ??z - jy, ] 



v = /ix + &y+/z + rx-?z, I (2). 



w = #x +/y + cz + ^y - ^x. J 



* " On the Theories of the Internal Friction of Fluids in Motion, &c." Camb. 

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



L, 3 



