443 
of Edinburgh, Session 1879-80. 
held with the bend down. In this position it constitutes a differ- 
ential thermometer of exceedingly high sensibility, founded on the 
difference of sulphurous acid steam-pressure due to difference of 
pressure in the two branches. One very remarkable and interesting 
feature is the exceeding sluggishness with which the liquid finds 
its level in the two branches when the external temperature is 
absolutely uniform all round. In this respect it presents a most 
remarkable contrast with a U tube, in other respects similar, but 
occupied by water and water-steam instead of sulphurous acid and 
sulphurous-acid-steam. If the U tube of water be suddenly inclined 
10 or 20 degrees to the vertical in the plane of the two branches, 
the water oscillates before it settles with the free surfaces in the two 
branches at the same level. When the same is done to the U tube 
of sulphurous acid, it seems to take no notice of gravity ; but in the 
course of several minutes it is seen that the liquid is sinking slowly 
in one branch and rising in the other towards identity of level. 
The reason is obvious. 
3. Vibrations of a Columnar Vortex. 
By Sir William Thomson. 
This is a case of fluid motion, in which the stream lines are ap- 
proximately circles, with their centres in one line (the axis of the 
vortex) and the velocities approximately constant, and approximately 
equal at equal distances from the axis. As a preliminary to treating 
it, it is convenient to express the equations of motion of a homo- 
geneous incompressible inviscid fluid (the description of fluid to 
which the present investigation is confined) in terms of ‘‘columnar 
co-ordinates” r , 6 , z, that is co-ordinates such that r cos 6 — x, 
r sin 6 = y. 
If we call the density unity, and if we denote by x, y, z the 
velocity-components of the fluid particle which at time t is passing 
through the point fx, y, z) : and by ~ , ~ ~ differentiations 
at ax ay az 
respectively on the supposition of x , y , z constant, t , y , z constant, 
t , x , z constant, and t, x, y constant, the ordinary equations of motion 
are 
3 H 
VOL. X. 
