95 
If the vortex filament is straight we have in this case 
V p = 0, which is an important condition in the case of fluid 
friction. On this account we should learn in whafc cases Vp 
moves with the fluid. 
Writing g = livQi + l&fa + 4v</> 3 as in (8), we can find that 
D*Vg = ^(vDA-V^i +■ vWaV^ + V&A Vfc) “ (SVgV)o' + Aj? .Sv<r 
= 2YvD,g - (Sv2 v)<r. + Sv<T.^ by (4). 
and therefore the condition that gYg may move with the 
fluid is that VvDJ^. = 0. 
7. The general case of motion of a viscous fluid is not 
capable of being simplified by the method of this paper, as 
no convenient expression can in general be found for Vp. 
As, however, there are cases in which simplifications can be 
made, I shall consider some of these cases. 
In the first case I shall suppose the fluid incompressible, 
as it is convenient to take these cases separately; and also 
suppose the filaments to be straight and parallel at right 
angles to the plane of motion. 
The equation of motion is 
vD t P + D t S 1 v0i + D f 2 2 v0 2 + vV + ^Vp = 0 
From this equation we will now suppose all terms of the 
form v V removed, so that we remain with 
D.Si V0i + D*2 2 V0 3 + ^Vp = O 
But p = S 3 a = 2 3 Ta . V0 3 = 2 V^ 3 
when v (/> 3 is now a unit vector perpendicular to the plane 
of motion. 
•*• Vp = Vv2.V03* 
If the filament be such that Tp is a maximum or mini- 
mum along it, we have shown that vp = ^ we must there- 
fore have DjS = 0, or the nature of the motion is not affected 
by the viscosity. 
