THE 



LONDON, EDINBURGH AND DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE, 



SUPPLEiMENT TO VOL. XVII. THIRD SERIES. 



LXXII. Remarks on Professor Challis's Investigation of the 

 Motion of a Small Sphere vibrating in a Resisting Medium. 

 By GEORGE BIUDELL AIRY, Esq. M.A. t F.R.S., Astronomer 

 Royal. 



To the Editors of the Philosophical Magazine and Journal. 



GENTLEMEN, 



IN your Number for December there is a paper by my 

 friend Professor Challis, on the theoretical resistance to 

 the motion of a sphere vibrating in an elastic medium. The 

 problem is so difficult, and so important in its application to 

 geodesy, and therefore of such general interest, that I have 

 thought it best to state, in a public communication, the diffi- 

 culty which I feel with regard to one step of the investigation, 

 and to request Professor Challis to remove my difficulty by 

 communication to your journal. 



I see nothing liable to objection in pages 463 and 464 ; 

 but with the top of page 465 my difficulty commences. The 

 differential equation, tacitly used by Professor Chaliis, is that 



f'(r at) f(r at) , . , 

 whose solution is v = s ; ~ 5 ^ - ; which equation 



is perfectly correct for waves, diverging with equal intensity 

 and with corresponding phase, in all directions from a centre ; 

 or, if not in all directions, it is yet true if the waves diverge 

 with equal intensity and corresponding phase through all the 

 angular directions included in a spherical sector bounded by 

 material planes, which (produced if necessary) would meet at 

 the centre of the sphere. But it is not true in any other case. 

 Thus we may have two such spherical sectors, separated only 

 by a material partition, and with waves of different intensities 

 and non-corresponding phases propagated in the two sectors, 

 from the centre, or from the surface of a small concentric 

 sphere ; and the equation applies to each sector separately ; 

 but if the partition be removed it no ionger applies to the 

 whole compounded sector. For the pressures against the par- 

 tition, produced by the fluids on the two sides, were different; 

 and, therefore, on removing the partition, a new motion of a 

 Phil. Mag. S. 3, Vol, 1 7. No. 1 1 3. Suppl. Jan. 1841. 21 



