91 
we have further that S v p = 0. 
T 
and may deduce that = 0 
To investigate another case, let us suppose that Tp in- 
creases or decreases in every direction except that of p. The 
condition for this is that Sp3v.p 2 = 0 where £ is any vector 
whatever. 
But Sp SpB Vp = .Sp 2 c> vp — SpV.Vpc) v-p 
— Sp 2 o v p — Sp 2 v p + SpS Sp v p 
= Sc)p 2 v p — S^p Sp v p. 
And this will not be zero for all directions of S unless p 2 V p 
= YpSp vp or unless Vp Vp 2 = 2.pSp Vp = 0 * 
This condition would have to hold if the vortex motion 
existed only in an indefinitely thin filament, and was either 
a maximum or a minimum at some point within it. 
Another case : suppose that the surface </>i contains vortex 
lines, and that Tp increases or diminishes as the surface is 
crossed on either side 
Then SpS v^iVp = 0. 
But SpS v0iVp = Sv0iSp vp + SY.VpV0i. V.p 
= Sv0]SpVp + SVpV0i. Vp 
= Sv0i(Spvp- Vpvp) 
= JSv«/)iVp 2 =0 
This condition must be satisfied at points in a vortex 
sheet or at points in a thin vortex filament if it contains a 
thread of fluid not affected by the vortex motion, or if the 
vortex motion is a minimum within. 
8. It has been shown (Section II.) that <7 may be written 
V P + V v 0i + S 2 V 02 + 21 3 v 08, but the whole portion Si v </>i 
+ S 2 v</> 2 + 2 3 v 0 3 will not generally contribute towards pro- 
ducing rotation ; for supposing Si = ~ we may write this 
cetyl 
portion as vQ + S 2 vf + 2 3 v fo, or generally we may write 
<r = v P + £, where £ = S 2 v </> 2 + S 3 v p 8 . In this we have made no 
* Notes on some Quaternion transformations. 
