On two Pulsating Spheres in a Liquid* 113 



series m is a whole number, whereas in the odd series it is a 

 whole number and a half, thus corresponding to the well- 

 known difference between elements of even series on the one 

 hand and odd series on the other. 

 Again, in the arithmetical series, 



; 2J ; 5 ; 8| ; 12 ; 15^ ; 19 ; 22J ; 26 ; 29± ; 33 



the common difference in the case of the first three members 

 is only 2^-, whereas after that point it is 3^ ; this corresponds 

 to the statement of Mendeljeff that the 2nd and 3rd series 

 (i. e. Li and Na series) of elements are more or less excep- 

 tional in their character and not strictly comparable with the 

 subsequent series. 



IX. On two Pulsating Spheres in a Liquid. 

 By A. L. Selby, M.A., Fellow of Merton College, Oxford*. 



A SPHERE is said to pulsate when it periodically dilates 

 and contracts. 



If two pulsating spheres are immersed in a liquid, in- 

 equalities of pressure in the liquid arise, and there is generally 

 a resultant pressure on each sphere urging it towards or away 

 from the other. 



An approximate expression of this is given in Basset's 

 ' Hydrodynamics ' (vol. i. p. 255), but the proof is rather long, 

 involving an integration of pressures over the surface of a 

 sphere. 



By a somewhat different method 1 have obtained in § 5 an 

 expression which may be carried further than Mr. Basset's 

 without much labour ; and in § 6 I have applied the method 

 to solve the more general problem relating to the case when 

 the radial velocities on the surfaces of the spheres are Zonal 

 Harmonics of the mth and nth degrees respectively, with the 

 line of centres as their axis. 



A still more general solution can be derived from this ; for 

 when the radial velocities on the spheres are any functions of 

 the distance from the line of centres, each can be expressed as 

 a series of Zonal Harmonics. 



Since I wrote this paper I have found one by Prof. Pearson 

 (Trans. Camb. Phil. Soc. vol. xiv. part ii.) in which this 

 problem is investigated ; but numerical values of the constants 

 involved are only given for Harmonics of low degree. 



In dealing with these questions some propositions on 



* Communicated by the Author. 

 Phil. Mag. S. 5. Vol. 29. No. 176. Jan. 18 ( U). I 



