94 
Thus the angle between p and p-x SU pvp is a;T VpSUp v p 
Tp^ 
whence the vector of curvature is 
Tp 3 
in which we may substitute from the formula, 
Vp 2 = 2V Vpp + 2(Sp v)p. 
This vector will give us the curvature at any point and 
also the torsional movement at that point during the mo- 
tion. It however contains vp, and can only be applied in 
special cases where the expressions can be simplified. 
In order to express the torsional motion of a vortex line 
at any point we will write Up = pi and the vector of curva- 
ture as e. Then e = YpiSpiV.pi (11) 
Then YeD 4 e will be a vector denoting the rate of torsion of 
the vortex line at the point considered during the motion. 
.*. YeD t e = Ye{ YD^piSp! V.pi + piSpi V .D*pi) 
= Spi v .piSelhpi - piSeSpi V .Ibpi. 
This vector, of course, vanishes if D f p is parallel to p ; a 
case which has been considered. 
The general case in which it vanishes is when 
e = YD,pSpV.D,p : 
in all other cases there will be a torsional flexure at each 
point in the vortex line. 
An expression for the radius of torsion of the vortex line 
at any time could also be found, but it is more complex 
than that already found, and would be of doubtful use in 
any case of application. We have already introduced two 
quantities Sp v p and v p for which we have found no proper 
expression, and of which the second is not easily interpret- 
able. 
In the case considered above when Tp decreases or in- 
creases in every direction round the vortex line, we have 
the simpler expression that 
that Sep = 0. 
e = 
Vp 
Tp’ 
with the condition 
