99 
of Edinburgh, Session 1866 - 67 . 
simple vortex ring is, that it always moves, relatively to the dis- 
tant parts of the fluid, in a direction perpendicular to its plane 
towards the side towards which the rotatory motion carries the 
inner parts of the ring. The determination of the velocity of this 
motion, even approximately for rings of which the sectional radius 
is small in comparison with the radius of the circular axis, has pre- 
sented mathematical difficulties which have not yet been overcome. 
In the smoke-rings which have been actually observed, it seems 
to be always something smaller than the velocity of the fluid 
along the straight axis through the centre of the ring ; for the 
observer standing beside the line of motion of the ring, sees, 
as its plane passes through the position of his eye, a convex* 
* The diagram represents precisely the convex outline referred to, and the 
lines of motion of the interior'fluid carried along by the vortex, for the case 
of a double vortex consisting of two infinitely long, parallel straight vortices 
of equal rotations in opposite directions. The curves have been drawn by 
