92 
special choice of the surfaces 0, hut since p moves with the 
fluid in such a way that if the two surfaces 0 2 and 0 3 inter- 
sect in vortex lines at any time, they will continue to inter- 
sect in the same lines, we may choose the surfaces 0 2 and 
(j ) 3 so as to intersect in vortex lines, and therefore put S 2 = 0, 
S 3 = 0 and p = 2ia. Whence ■~ L = 0, or, the strength of the 
vortex is constant as we pass along the filament — a pro- 
position generally proved by the aid of Green’s Theorem. 
(TO 
Also S£p = 0 and So-p = ^Sct v P = - Hs 13 — . 
The first of these equations shows that the axis of rota- 
tion is perpendicular to the part of the velocity essential to 
the rotation. 
4. Or we might by a proper choice of the surfaces fa ex- 
press g merely as 2 iV(j>i , and since D*2 = 0, this would 
remain as the proper form of o\ This form of <r expressed 
as v P + 2iV0i seems that most applicable to the usual Car- 
tesian notation ; comparing it with the usual form for irro- 
tational motion we see that two new scalars are introduced, 
connected by the two scalar equations = 0, D^i = 0. 
or 2i - SiS V 0i V 2i = S V P V 2i 
and ?»i-2i(vf) 2 = SvPV(/)i 
These two equations show very clearly that with a certain 
definite system of vortex motion a definite irrotational 
velocity must exist accompanying it, and that if any ex- 
traneous velocity could be imposed upon the system, the 
velocity must in some way be modified, and react upon the 
disturbing velocity. 
The fact that o- may be written as vP + ^v^i seems to 
me sufficiently important to make me introduce the theorem 
in Pure Mathematics on which it depends : namely, that 
the ordinary differential equation P dx + Q dy + R dz consists 
