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faces k and \p coincide wliere vortex motion does not exist, 
for the = 0 at all points, and the normals to the 
surfaces at all points are parallel. Whence the surfaces 
/:: = const, and ^ = const, coincide except at points where 
vortex motion exists. In order therefore to apply this 
method to the solution of rotational motion of a fluid, we 
should consider k'^ip an additive term, and take for k and xp 
values such as to make the surfaces k = const, and xp = const, 
move with the rotational fluid, and always intersect in vor- 
tex lines, while the term v0 would be taken to satisfy 
the general irrotational motion. 
5. To investigate the energy within a surface drawn 
within a fluid we get by continually using a modification of 
Green s Theorem and omitting Thomson’s correction 
2T = y " adv = y' S (A(/> + k\;xp) (xdv = J Sa^(j)dv + J Sktryif^dv 
= — J'S'^(T(f)dv + J SffipvciS 
— / S\/ka'xpdv + f Sk\pirrdS 
- - J Svo-(0 + k^p)dv - J Sxp\7ka-dv + J S{(j) + kxp)<ryd^ 
= -ff^^kadv + fs{cl) + k-p)(TpdS V. 
where v stands for the unit normal to the surface, and where 
the last term vanishes if there is no flow across the bound- 
ing surface. 
We may then use the equation of continuity to give 
other forms to the volume integral. Thus 
SxpykfT = il^Syk(y^ + 
= pS(\7Jc\7(j) - k\7^(j) - k\^p) 
forms which will allow us to use Green’s Theorem again. 
From the rate of change of the circulation in a closed - 
circuit moving with the fluid we get 
D^y ' crdr = 0 or J'Dtffdr + J ad.T)f = y'D^o’^r + J (rS(r = 0. 
The second is zero over every closed circuit. In order that 
the first may be so we must have Vv-Dto- = 0, or referring to 
V{vD<^.V+ + =0 VI. 
a further justification for our assumption. 
