844 Dr. T. Craig on Steady Motion in 



these equations can now be transformed into the following: — 



O. <X 



cTP 



L .... (5) 



where r= - is the "kinematic coefficient of viscosity," and 



P-j^+V + tf. 



For ^ = 0, these become the equations given by Lamb in the 

 ' Proceedings ' of the London Mathematical Society. In the 

 case of /x = 0, we know (vide Lamb's ' Treatise on Fluid Motion/ 

 page 173) 



"that dP dP dP ' 



11- h V ~ 7 \-W-t- = 0, 



da; dy dz 



dP dV dP „ 



" so that the surfaces 



P= constant 



" contain both stream- and vortex-lines. Further, denoting by 

 dn an element of the normal to such a surface at any point, 

 we have 



dP . a 



-j- = qco sin u ; 



where co is the spin, and 6 is the angle between the stream- 

 line and vortex-line at the point where the normal is drawn. 



" The conditions, then, that a given state of motion of a per- 

 fect fluid may be a possible state of steady motion are as fol- 

 lows: — It must be possible to draw in the fluid an infinite 

 number of surfaces each of which is covered by a network of 

 stream-lines and vortex-lines; and the product ga> sin 6 dn 

 must be constant over each such surface, dn being the length 

 of the normal drawn to a consecutive surface of the system." 



For the case of a viscous fluid, the reductions are as simple as 

 those which lead to the above results. To the. equations of 

 motion (5) it is of course necessary to add the equation of con- 

 tinuity, 



du dv dw 



dx dy dz ' 



