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TRANSACTIONS OF THE SECTIONS. 7 
On the Perfect Biackness of the Centre of Newton’s Rings. 
By G. G. Sroxes, M.A. 
The absence of all reflected light at the centre of Newton’s rings, when formed 
between two lenses of the same substance, was explained long ago by Fresnel, by 
the aid of a law discovered experimentally by M. Arago, that light is reflected in 
the same proportion at the first and second surfaces of a transparent plate bounded 
by parallel surfaces. It occurred to the author that this law might be obtained very 
simply from theory, by means of what may be called the principle of reversion. By 
this is meant the general dynamical principle, that if in any material system, in 
which the forces depend only on the positions of the particles, the velocity of each 
particle be suddenly reversed, the previous motion will be repeated in the reverse 
direction. It follows from this principle, that in the case of a series of waves of 
light incident on the surface of an ordinary medium, and producing a series of re- 
flected, and a series of refracted waves, if the vibrations in the reflected and refracted 
series be reversed, the incident series will be produced, only in a reverse direction. 
But the reflected and refracted series, when reversed, would each produce a series 
of reflected, and a series of refracted waves; and it follows from the principle of 
superposition, that, of these four series, the two which are situated within the medium 
must neutralize each other, and the two which are situated outside of the medium 
must together produce the incident series reversed. Two equations are thus ob- 
tained, whereby the perfect blackness of the centre of Newton’s rings is explained. 
These equations are those which are written in Airy’s Tract b=—e, cf=1—e&. 
The detail of this method will soon appear in the Cambridge and Dublin Mathe- 
matical Journal. 
On the Resistance of the Air to Pendulums. By G.G. Stoxss, M.A. 
There are a few cases in which the resistance of a fluid to a pendulum oscillating 
in it have been calculated on the common theory of hydrodynamics, in which the 
pressure is supposed equal in all directions. The results in the cases of a sphere and 
of a long cylindrical rod are very simple, and may be expressed by saying that the 
mass of the pendulum must be conceived to be increased in the former case by 
the half, and in the latter by the whole of the fluid displaced ; this additional mass 
increasing the inertia of the pendulum without increasing its weight. These results 
agree very nearly with Dubuat’s experiments on spheres oscillating in air, Dubuat 
having employed spheres of large diameter, and with Baily’s experiments on cylin- 
drical tubes of 14 inch diameter. With smaller spheres and thinner rods, however, 
the results no longer agree with experiment, as appears from the experiments of 
Bessel and Baily, the discrepancy being so much the greater as the diameter of the 
sphere or rod is the smaller. The author stated that he had solved the problem 
in the cases of a sphere and of a cylindrical rod, using instead of the common 
equations of hydrodynamics the equations which he had given in the 8th volume | 
of the Cambridge Philosophical Transactions, and which had been previously ob- 
tained, by different methods, by Navier, by Poisson, and by M. de Saint-Venant. 
The résult in the case of the sphere is very simple, although the function which 
expresses the state of motion of each particle of the fluid is rather complicated. It 
appears that the effect on the time of oscillation will be obtained by conceiving a 
mass equal to(— BL + ) m to be added to the mass of the sphere, m being the 
3 1 
mass of fluid displaced, and s =F eee 
Tr 
time of oscillation, @ the density, and « the constant so denoted in the author’s 
paper referred to above. This constant must be determined by experiment for each 
fluid in particular, and even for the same fluid at different temperatures, since it 
probably decreases as the temperature increases. Besides the effect on the time of 
oscillation, the are of oscillation slowly decreases, the expression for the decrement . 
, @ being the radius of the sphere, ¢ the 
of the arc involving the quantity 2 s(1+s)m. 
The result in the case of the cylinder is much more complicated, and requires the 
_ numerical calculation of functions belonging to this particular problem. The author 
