By Huntington E.V.
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Additional resources for An Improved Equal-Frequency Map of the Normal Correlation Surface, Using Circles Instead of Ellipses
9 , 1 find that when a ball pendulum oscillates in a concentric spherical envelope, the effect of the b3 + 2a3 inertia of the fluid is to increase the mass of the ball by ^-TTS §r times that of the fluid displaced, a being the radius of the ball, b that of the envelope. Poisson, in his solution of the problem of the sphere, arrives at the strange result that the envelope does not at all retard the oscillating sphere. I have pointed out the erroneous step by which he was led to this conclusion, which I am clearly called upon to do, in venturing to differ from so high an authority.
If the fluid were infinite we should have for the motion corresponding to that of the given sphere Co? ^" The motion reflected from the plane is evidently the same as that corresponding to the motion of the imaginary sphere in an infinite mass of fluid, for which we have , Ca3 cos#' Now r cos 0' — r cos 0, r' sin ff sin at' = r sin 0 sin co, r sin & cos m' + r sin 0 cos o> = 2h; r1 = r2 + W — 4
hr sin 0 cos &>}f ' Retaining only the terms of the order asr/h3 or r4/^3, so as to get the value of d
Sin0 cos (o+Osin 6 sin w, we easily find, by making use of rectangular co-ordinates, changing the direction of the axes, and then again adopting polar coordinates, that the above term in <£ takes the form D cos 9l. r"z, ! being measured from some line passing through the origin. The motion will therefore be the same as that round a ball pendulum in an incompressible fluid, the centre of the ball being in the origin; a case of motion which will be considered immediately. In order to represent the motion at different times, ON SOME CASES OF FLUID MOTION.