8 REPORT—1848. 
has expressed the effect on the time and arc of oscillation by means of ascending 
series, which are always convergent. The result involves the remarkable trans- 
cendent I’ (+) , I’ (m) being the derivative of I (wm). The author has also ob- 
tained a descending series, which is much more convenient for numerical calcula- 
tion when the diameter of the cylinder is large. It appears from theory that the 
factor which Baily has denoted by m increases indefinitely as the radius of the 
cylinder decreases: the mass of air which must be conceived to be dragged by the 
cylinder decreases, but very slowly, varying ultimately as the square of the reci- 
procal of the logarithm of a quantity which varies as the time of oscillation divided 
by the square of the radius of the cylinder. 
The diameters of the cylindrical rods employed by Baily were °410, *185 and 
‘072 inch; and the corresponding values of n, which according to the common 
theory of hydrodynamics ought to be equal to 2, were 2°932, 4°083, and 7°530. 
Each of these results furnishes an equation for the determination of #, or rather of 
re which is what enters into the calculation ; and the three results concurred 
in eiving to the latter quantity a value lying between ‘1] and ‘12, an inch and a 
second being the units of space and time. The value ‘11 satisfied very nearly the 
experiments on spheres suspended by fine wires; the effect of the wire, which on 
this theory is not quite insensible, being taken into account, and a small correction, 
estimated at half the correction calculated for a spherical envelope of the same 
radius, being made for the finite size of the hollow cylinder within which the spheres 
were swung. 
On the Equilibrium of Magnetic or Diamagnetic Bodies of any Form, under 
the Influence of the Terrestrial Magnetie Force. By Prof.W.Tuomson. 
If a body composed of a magnetic substance, such as soft iron, or of a diamag- 
netic substance, be supported by its centre of gravity, the effects of the terrestrial 
magnetic force in producing magnetism by induction, and in acting on the magnetism 
so developed, are in general such as to impress a certain directive tendency on the 
mass. The investigation of these circumstances leads to results according to which 
the conditions of equilibrium of a body of any irregular form may be expressed in a 
very elegant and simple manner. 
In the present communication I shall merely give a brief general explanation of 
the conclusions at which I have arrived, without attempting to state fully the pro- 
cess of reasoning on which they are founded, as this could not be rendered intelli- 
gible without entering upon mathematical details, which must be reserved for a 
paper of greater length. 
In the first place, if the body considered be an ellipsoid of homogeneous matter, 
supported by its centre of gravity, it is clear that it will be in equilibrium with any 
one of its three principal axes in the direction of the lines of force; and if it be put 
into any other position, the action of the earth upon it will be a couple, of which 
the moment may be expressed very simply in terms of the quantities denoting the 
position of the axes with reference to the direction of the terrestrial force, and cer- 
a} constant magnetic elements depending on the substance and dimensions of the 
ody. ‘ 
In considering the corresponding problem for a body of any irregular form, we 
readily obtain for the components of the directive couple, round three rectangular 
axes chosen arbitrarily in the body, expressions involving nine constant magnetic 
elements. I have succeeded in proving that six of these elements must be equal, 
two and two; so that the entire number of independent constants is reduced to six. 
I have thus arrived at the interesting theorem, that there are, in any irregular body, 
three principal magnetic axes at right angles to one another, such that if the body be 
supported by its centre of gravity, it will be in equilibrium with any one of these 
axes in the direction of the terrestrial magnetic force. If the body be held in any 
other position, there will be a directive couple of which the moment is expressible 
in precisely the same form as in the case of an ellipsoid, in terms of the magnetic 
elements of the body, and of the quantities denoting its position. 
