1890.] On Stokes s Current Function. 49 



and the expression (1), and consequently also (2), satisfies the 

 differential equation 



J2 V , &y I ty 



~; o ~r T~5 -3 w ....................... \' J Ji 



cftr* az~ w az 



or, as I shall write it, D^- = 0. 



When the motion is rotational, (3) no longer holds. In fact, as is 

 well known, we have under all circumstances 



D^= -2o>, 



" 



where w is the resultant molecular rotation at the point (zzr, z). 

 Thus, if there is molecular rotation in the fluid, (3) is replaced by 



a.3 



.. d 2 d 1 rf 1 d 2 



Again, it V" stand for the operator + _| _| -- 



rf'ar' 5 az - ar a^nr -nr" a0- 



</6 being the azimuthal angle about the axis of symmetry, it may be 

 seen at once that 



sin 

 Consequently (3a) may be written 



(4). 



sin sin 



- = 2trwX - - 



-ar 

 ^ 2o> sin 



Consequently 



sin <& u) dx dy dz 



where T^O ^ s a solution of (3). 



Or \(f consists of a solution of (3) together with - X the poten- 



2?r sin 



tial at the point considered of a distribution of mass of density at 

 any point sin X the molecular rotation at that point. This result 

 is given by Basset, ' Hydrodynamics,' vol. 2, 306. 

 I give one other general result. ?ince 



the circulation in any evanescible circuit drawn in a meridional 

 plane is 



- 1 1 D f dw fh ................. (5), 



VOL. XL1X. 



