﻿Sir W. Thomson on Approach caused by Vibration. 427 



interesting as a mathematical problem, which will no doubt 

 prove tractable under certain assumed conditions as to tension 

 or rigidity of the containing film. 



Yours very truly, 

 Professor Guthrie. William Thomson. 



Nov. 24, 1870. 



Dear Sir, — My letter to you of yesterday requires some limi- 

 tation. 



Even a perfectly rigid solid less dense than the surrounding 

 medium must, if small enough, exhibit the repulsion which you 

 tell me Shellbach has observed for balloons of hydrogen. On 

 the other hand, if large enough, it is clear that, as I said yester- 

 day, a rigid solid must exhibit attraction ; for if very large 

 relatively to the vibration, it cannot move sensibly as a whole to 

 and fro with each vibration. Now, if the rigid body is held ab- 

 solutely fixed, it follows from your own experiments and reason- 

 ing, as well as from the mathematical theory, that the vibrator 

 will experience attraction, irrespectively of the density of the 

 fixed rigid solid ; and therefore when the vibrator is attached to 

 a fixed stand and the rigid solid left free to move, the latter, if 

 large enough, must experience repulsion. Consider, first, any 

 very small portion of the fluid moving to and fro in virtue of 

 the periodic motion sustained throughout the fluid by the vi- 

 brator. Let it be so small in all its dimensions that it experi- 

 ences no sensible change of shape during its period of motion. 

 If this portion becomes rigid its rigidity will but infinitesimally 

 affect its own motion and that of the fluid surrounding it. If, 

 now, by a proper application of force varying periodically ac- 

 cording to the simple harmonic law, the little solid be caused to 

 move to and fro faster than the neighbouring fluid or slower 

 than the neighbouring fluid, it will require a certain constant 

 average force to prevent its mean position from yielding in a 

 certain direction from or towards regions of greater fluid motion. 

 When the superimposed vibration augments the natural vibra- 

 tion of the solid, the direction in which it tends to move is from 

 regions of greater towards regions of less fluid vibration; in 

 the other case the contrary. Lastly, the imagined differential 

 vibration becomes a reality if the density of the little solid is 

 either greater or less than that of the fluid. When it is less 

 than that of the fluid, the differential motion augments the vi- 

 bration ; when the density is greater, the differential motion 

 diminishes the vibration. Hence an infinitesimal solid of dif- 

 ferent density from the surrounding fluid tends towards the re- 



I wrote the letter in the text, render it, I think, quite certain that the ex- 

 planation here suggested is correct. — W. T. 



