212 
T) j. /ci \ ’ d ' d ' d 
But + + 
' ,d a — 
^ • a T\' T\' y 
and the condition for steady motion isDi0 = O, showing a 
close resemblance in form between the velocity in this case 
and the rotation in a perfect fluid. 
We may at any time, by elimination of t, find two sur- 
faces of the nature </> for which (^ = 0, and since in steady 
motion = 0, the property will continue in the surface. 
Taking the two such surfaces for </>2 and ^g, they will act as 
fixed boundaries, their intersections will be the stream lines 
and the form of (j will be 
(T 
a. 
H" ; 
In spite of the simplicity of this form, it does not appear to 
yield a convenient form of condition affecting Difr, nor 
for Vvo". 
The following property can, however, be deduced. In a 
perfect fluid under conservative forces, we must have (Sn-y) 
of the form vP 
But (So'V)o’ = + Vo-VV'T. 
V.o-Y^cr = vQ_say, 
^iSgV^2 + — vQ 
or 
dQ 
whence ^ = 0, and and ^^become 0 when acted on by 
d(j)i d(f)^ 
Dt, whence Q is a function of ^2 and ^3 only. The existence 
of this surface Q, on which both stream lines and vortex 
lines lie, is dependent on the existence of vortex motion, 
but if the surface exists we may take if in place of the 
surface ^3 to indicate the stream lines, and then we get Sg = 0 
and 01^2 = 1 , or 22 = ' ’ rotation would be given by 
01 
1 
Vv<t = Si.Vv</>2vQ + --YvQv0i (3) 
01 ^ ^ 
The cases of steady vortex motion will, I think, generally 
occur in cases allied to the surfaces of discontinuity inves-’ 
tigated by Helmholtz, and the surface Q will then be such 
a surface. 
