207 
This form enables us to express in a general way the crite- 
rion that the vortex motion may exist in the form which it 
will be seen to take in a perfect fluid. Taking the two 
other similar quantities, and equating each to zero, we get 
YvDto- = b or D^rr of the form v?- 
This notation also enables us to give a verbal expression 
for V V 0-. Thus let PQB- lie on a line of intersection of two 
surfaces 02 ^^nd 0g, and let them be distinguished by 0i, 
01 + and 01 - Then the vector difference of the 
2 i ^ 
velocities at Q and R is -^^0i, and re- 
solving parallel to the face 0i of the 
element, having P for centre, these diffe- 
rential velocities on the parallel faces 0i, 
we get 
^0iV^" U V 01 X area = -f~x area V 
.d(T . 
d(p\ 
K 
— volume X V 0i 
d(f)i ^ 
da 
d(pi 
V</»i 
and similarly for the other faces. 
volume X V V AT = resultant differential velocity and V v o- = 
mean resultant differential velocity per unit of volume as 
the result of three shears and rotations, and by a similar 
method we may find the force due to viscosity, arising in 
consequence of the motion noticed above, upon the faces 0i, 
being due to the rate of change of the differential velocities 
just found. It will plainly be proportional to volume 
II. 
The equation of motion is on the generally adopted 
theory 
Dto-v + vY ]o vSvo- + -^vV = 0 (1) 
^9 9 
where V Y represents what would be written in Cartesian, 
- 1 {Xlx + Y<V + Zlz) + p. ) 
