115 
That is, the axes of vortex motion perpendicular to the 
lines of flow. A case satisfying the conditions of parallel 
cylindrical vortices, and circular vortex rings. 
If (T-V0 integrahle by a factor, the condition is 
S(<r - V0) V(o’ - V</>) = 0 or S(o- - V0)p = 0. 
From this scalar equation cj) could be found, and we may 
consider o- = v0 + a perfectly general form of expression 
for the velocity at any point. 
3. The kinematical condition that a must satisfy is the 
“equation of continuity,” which if the fluid is incompres- 
sible takes the form Svo- = 0 or 
+ + ^ = 0 II. 
and the angular velocity at any point in the fluid is given by 
2p = Y „III. 
The form taken by the equation of motion, when this 
expression is substituted for cr, will now be found ; for 
= y ^ may be written 
U,(v^> + *V>/') = v(v+|-^ 
( \ 
Now D(V = V-I^t “ where A acts only on the cr in or 
d^ - (So-v) ; therefore 
Dj.v</> + = V. ^4 + ^V- + A { (So-v)<^ + Ic (So-v)v/^} 
= V. V- J 
and finally the equation of motion is 
D,vD,.^ + D,fev>;' + /CV.D,1^' + = vfv + £j 
Now this equation is equivalent to three scalar equations 
and if we make in it the substitutions D/c = 0 D^xp = 0, we 
reduce it to a single equation and solvable ; for then 
vD,^=v(v + £-^) 
^ p (T , 
or = V + - - 7T + V 0 
^ m 2 
