CHAPTER III. 



IRROTATIONAL MOTION. 



38. THE present chapter is devoted mainly to an exposition 

 of some general theorems relating to the class of motions already 

 considered in Arts. 22 27; 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, 

 those at an infinitely near point (x + X, y + F, z + Z) are 

 du v du v du ^ 



~T -A- I 7 -* ~T~ ~~T~ 



dx dy dz 

 dv , dv , dv r 



dx 



dy 



If we write 



du , dv dw 



a= -J- , o = ^-, C = -T- , 

 dx dy dz 



dw dv 



dw dv 



du dw 



du dw 



dv du 



dv du 



equations (1) may be written 



* Camb. Phil. Trans. Vol. vin., 1845. 



