90 
Thus if (Spv)<r = 0, a case which includes that of straight 
parallel vortex filaments, 
then "D*g = 0 , and by (4) g may he written 
V (pi + 12 V 02 4- l 3 V 03 where = 0. 
In this case by ( 6 ) we have SpS£v< 7=0 or Spt>^ = 0, or the 
relative motion is perpendicular to p. 
Also by (7) D,(s^) = S^; =SvD,P.^; and r»,S^S = 0, 
and v g moves with the fluid as g does. 
A more general case, namely, when (Sp v )<r 2 = 0 gives very 
similar results. It may he written StrSpv <7 = 0 or Sod\o = 0, 
and is the case when the tensor of the velocity is constant 
along a vortex line, including the case of an ordinary vor- 
tex ring. 
We deduce by ( 6 ) that SpS< 7 V<r = 0 , 
and obtain 
D*(So-g^ = S 6 g as before. 
If So-Sp Vff = 0 at any point of a vortex ring, it denotes 
that at that point o - 2 is a maximum or a minimum. 
Another case in which we get some additional simplifica- 
tion is when the vortex line through any point moves 
parallel to itself. In this case 
VpD*p = 0 or VpSp v <r = 0 
where we may omit the H or not, as we please. 
Write as in (4) ^ = l x Vpi + h Vpz + h Vp 8 
Then because = 0, 
H H 
T) t l 2 D ^3 D t i 
<3= <!= «»= < gay 
v 1 ^2 ^3 v 
.*. l\ — Ihi , l 2 = Ik 3 , l 3 = lk 2 where D t k — 0 
and ^ = ?(* l v^i + iaV^s + ^»V^| 
2D,| = D^|^v 0 i + &o.} =D,log7.|. 
