209 
HS 
Vv^are--^ times the corresponding edge, and if the 
cp.1 
fluid is incompressible this ratio remains unaltered in time, 
and the manner of variation, if the fluid is compressible, is 
indicated by the formula. 
Also since = 0, if V v o- is ever = 0 it always was and 
will be = 0, and the strength of the section of the vortex 
S S — — 
on the surface is S v</ji.2ia = and is constant in 
all time. 
If we compare the form of o- here obtained with that 
assumed by Helmholtz, we find that his form is not suffi- 
ciently general ; since he writes 2i v 0i + 2-2 v + 2g v 03 = V w 
where w is a vector with the condition S v w = 0. That is, he 
includes two undetermined quantities instead of three, and 
does not obtain in a distinct form the essential condition, 
nam ely = 0, affecting them. 
The quantities 2 are in no way dependent on finite forces 
which are acting (under the hypothesis), but entirely on the 
initial conditions (boundary conditions not now being 
entertained). 
Unless the density be a function of the pressure only, the 
relations just proved will certainly not hold, for then 
DifT + V V + i = 0 , 
Ki = + 2i where 2i = /(0i. 02-03 and (). 
d<pi 
in which case the fluid pressure will of themselves cause 
vortex motion. 
This seems to be a quite possible condition in a gas 
rapidly heated or expanded. 
Now turning to the terms containing fx and imagining p 
a function of p only, and remembering that p is independent 
of p, then 
P 
