87 
The equation (6) shows a curious theorem in the kinematics 
of vortex motion, connecting the relative velocities of points 
near a vortex line. 
The equation may be written Sp(S«r8 v )cr = S£(S xp v )v where 
we shall consider, for the moment, that p and 8 are unit 
vectors — since we may divide by the product of their tensors. 
The theorem may be stated thus. If 0 be any point on a 
vortex line at which o- expresses the velocity, and if equal 
small distances (pc) be taken in the directions of p and of 
any vector 8; the velocities relative to that at O are 
-a(S pv)<r and -#(S£v)<r, and are such that their compo- 
nents resolved along 8 and p respectively are equal to one 
another. If the vector 8 be one which moves with the 
fluid (as g) we may then state the theorem of the actual 
relative motions, for we could write the equation 
S^D,S = S® ( £- 
Similarly |D*(T£) 2 = S£(S3v)<r = the projection of the relative 
velocity upon 8 itself; and - Va(Sa v)<r is the vector express- 
ing the double of the rate at which areas are described by 
the vector 8. If we operate upon this with D* we get 
-VS(S5v)lV 
If we project the double area found above on the plane 
perpendicular to p we get 
-Sp8(S8v)<r-fTp (9) 
But - S P 3( S3 V )(T = sa.psa V O- = S OSpV <r + SSVVpS V <7 
= S^VpC) v <7 — Sc)Sp() V <7 
= 0W-sspv<7 
Suppose, for convenience, that p, unless when distinctly 
derived from <y, stands for Up, a unit vector. 
Then - SaS^V a = Sp Vpffipv <r 
- - twice the rate of description of projected areas by the 
vector Vp8. And we have the theorem that double the 
sum of the areas described by the small vectors 8 and VUp8 
is Tp(T8) 2 sin 2 0, where 0 is the angle between 8 and p. It 
