100 Proceedings of the Royal Society 
outline of an atmosphere of smoke in front of the ring. This 
convex outline indicates the bounding surface between the quantity 
of smoke which is carried forward with the ring in its motion, 
and the surrounding air which yields to let it pass. It is not 
so easy to distinguish the corresponding convex outline behind 
the ring, because a confused trail of smoke is generally left in the 
rear. In a perfect fluid the bounding surface of the portion carried 
forward would necessarily be quite symmetrical on the anterior and 
posterior sides of the middle plane of the ring. The motion of the 
surrounding fluid must be precisely the same as it would be if the 
space within this surface were occupied by a smooth solid ; but in 
reality the air within it is in a state of rapid motion, circulating 
round the circular axis of the ring with increasing velocity on the 
circuits nearer and nearer to the ring itself. The circumstances of 
the actual motion may be imagined thus : — Let a solid column of 
india-rubber, of circular section, with a diameter small in propor- 
tion to its length, be bent into a circle, and its two ends properly 
spliced together, so that it may keep the circular shape when left 
to itself ; let the aperture of the ring be closed by an infinitely thin 
film; let an impulsive pressure be applied all over this film, of in- 
tensity so distributed as to produce the definite motion of the fluid, 
specified as follows, and instantly thereafter let the film be all 
liquefied. This motion is, in accordance with one of Helmholtz’s 
laws, to be along those curves which would be the lines of force, if, 
in place of the india-rubber circle, were substituted a ring electro- 
magnet,* and the velocities at different points are to be in pro- 
portion to the intensities of the magnetic forces in the corresponding 
points of the magnetic field. The motion, as has been long known, 
will fulfil this definition, and will continue fulfilling it, if the 
initiating velocities at every point of the film perpendicular to its own 
Mr D. M'Farlane, from calculations which he has performed by means of the 
equation of the system of curves, which is 
y l 
2x 
N + l 
a 'N 
_ _ + where log e N = — 
The proof will be given in the mathematical paper which the author intends 
to communicate in a short time to the Royal Society of Edinburgh. 
* That is to say, a circular conductor with a current of electricity main- 
tained circulating through it. 
