1895.] of a Viscous Incompressible Fluid. 315 



from which by integration we obtain 



^+V + ^q' + F(x) = const (6), 



r 



where ^ (%)=]/(%) ^%- 



We also have, from equations (3), 



Also, from equations (3), combined with the equation of con- 

 tinuity, we obtain 



|^>+.VMogr=0, 

 d {x, y) 



which may be written in the form 



1^^ + - {^" log/(%) - V^ log m] = (8). 



The motion is therefore given by the three equations (6), (7), 

 (8). These equations however fail to give a minimum theorem, 

 except of a very restricted kind. 



2. The particular case in which the vorticity is so distributed 

 that we have 



VMogr = 0, 



is worthy of note. In this case we have m a function of ;!^, and the 

 equations reduce to the same form as they assume in the case of 

 the perfect fluid, viz. 



r 



We also have 



2r=/;(%)- 



In the general case of the preceding article, if we write down 

 the variation of the expression 



//" 



\dxdy[\{ii^ + v')-F{x)], 

 and transform it according to the usual method, then equations (7) 



