ON RECENT RESEARCHES IN HYDRODYNAMICS. 15 
Ay 
_ finds that the sound suffers total internal reflection. The expression for the 
_ disturbance in the second medium involves an exponential with a negative 
_ index, and consequently the disturbance becomes quite insensible at a di- 
_ stance from the surface equal to a small multiple of the length of a wave. 
_ The phase of vibration of the reflected sound is also accelerated by a quan- 
tity depending on the angle of incidence. It is remarkable, that when the 
_ fluids considered are ordinary elastic fluids, or rather when they are such 
that equal condensations produce equal increments of pressure, the expres- 
sions for the intensity of the reflected sound, and for the acceleration of 
_ phase when the angle of incidence exceeds the critical angle, are the same 
as those given by Fresnel for light polarized in a plane perpendicular to the 
plane of incidence. 
VY. Not long after the publication of Poisson’s memoir on the simultaneous 
motions of a pendulum and of the surrounding air*, a paper by Mr. Green 
was read before the Royal Society of Edinburgh, which is entitled ‘ Re- 
searches on the Vibration of Pendulums in Fluid Media+.’ Mr. Green does 
not appear to have been at that time acquainted with Poisson’s memoir. The 
_ problem which he has considered is one of the same class as that treated by 
Poisson. Mr. Green has supposed the fluid to be incompressible, a suppo- 
_ sition, however, which will apply without sensible error to air, in considering 
_ motions of this sort. Poisson regarded the fluid as elastic, but in the end, in 
_ adapting his forthula to use, he has neglected as insensible the terms by 
which the effect of an elastic differs from that of an inelastic fluid. The 
_ problem considered by Mr. Green is, however, in one respect much more 
general than that solved by Poisson, since Mr. Green has supposed the oscil- 
lating body to be an ellipsoid, whereas Poisson considered only a sphere. 
_ Mr. Green has obtained a complete solution of the problem in the case in 
_ which the ellipsoid has a motion of translation only, or in which the small 
_ motion of the fluid due to its motion of rotation is neglected. The result is 
_ that the resistance of the fluid will be allowed for if we suppose the mass of 
the ellipsoid increased by a mass bearing a certain ratio to that of the fluid 
displaced. In the general case this ratio depends on three transcendental 
quantities, given by definite integrals. If, however, the ellipsoid oscillates in 
3 the direction of one of its principal axes, the ratio depends on one only of 
_ these transcendents. When the ellipsoid passes into a spheroid, the tran- 
_ scendents above-mentioned can be expressed by means of circular or loga- 
_ rithmic functions. When the spheroid becomes a sphere, Mr. Green’s result 
_ agrees with Poisson’s. It is worthy of remark, that Mr. Green’s formula will 
enable us to calculate the motion of an ellipse or oircle oscillating in a fluid, 
_ in a direction perpendicular to its plane, since a material ellipse or circle may 
_ be considered as a limiting form of an ellipsoid. In this case, however, the 
motion would probably have to be extremely small, in order that the formula 
should apply with accuracy. 
__ Ina paper ‘On the Motion of a small Sphere acted on by the Vibrations of 
_ an Elastic Medium,’ read before the Cambridge Philosophical Society in April 
_ 1841, Prof. Challis has considered the motion of a ball pendulum, retaining 
in his solution small quantities to the second order. The principles adopted 
by Prof. Challis in the solution of this problem are at variance with those of 
Poisson, and have given rise to a controversy between him and Mr. Airy, 
7... will be found in the 17th, 18th and 19th volumes of the Philosophical 
w+ 
Ton 
_____-* Mémoires de I’Académie des Sciences, tom. xi. p. 521. 
__ * This paper was read in December 1833, and is printed in the 13th volume of the So- 
_ ¢iety’s Transactions, p. 54, &c. i i 
___ ¢ Transactions of the Cambridge Philosophical Society, vol. vii. p. 333. 
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