135-137] MUTUAL INFLUENCE OF TWO SPHERES. 207 



where \xy'\ denotes the mean value of xy. If x, y differ in phase by less than a 

 quarter-period, this force is one of repulsion, if by more than a quarter-period 

 it is one of attraction. 



Next, let B perform small periodic oscillations, while A is held at rest. The 

 mean force which must be applied to A to prevent it from moving is 



where [j/ 2 ] denotes the mean square of the velocity of B. To the above order of 

 approximation dJV/dc is zero, but on reference to Art. 97 (xv) we find that 

 the most important term in it is - 127rpa 3 6 6 /c 7 , so that the force exerted on 

 A is attractive, and equal to 



tte^rW .............................. (vii). 



This result comes under a general principle enunciated by Lord Kelvin. 

 If we have two bodies immersed in a fluid, one of which (A) performs small 

 vibrations while the other (B) is held at rest, the fluid velocity at the surface 

 of B will on the whole be greater on the side nearer A than on that which is 

 more remote. Hence the average pressure on the former side will be less 

 than that on the latter, so that B will experience on the whole an attraction 

 towards A. As practical illustrations of this principle we may cite the 

 apparent attraction of a delicately-suspended card by a vibrating tuning-fork, 

 and other similar phenomena studied experimentally by Guthrie* and 

 explained in the above manner by Lord Kelvin f. 



Modification of Lagranges Equations in the case of 

 Cyclic Motion. 



137. We return to the investigation of Art. 134, with the 

 view of adapting it to the case where the fluid has cyclic irrota- 

 tional motion through channels in the moving solids, or (it may be) 

 in an enclosing vessel, independently of the motion of the solids 

 themselves. 



If K, K, tc", ..., be the circulations in the various independent 

 circuits which can be drawn in the space occupied by the fluid, the 

 velocity-potential will now be of the form 



where = g r i<k + &&+ ........................ (1), 



* "On Approach caused by Vibration," Proc. Roy. Soc., t. xix., (1869); Phil. 

 Mag.,Ko\. 1870. 



t Reprint of Papers on Electrostatics, &c., Art. 741. For references to further 

 investigations, both experimental and theoretical, by Bjerknes and others on the 

 mutual influence of oscillating spheres in a fluid, see Hicks, "Keport on Eecent 

 ^Researches in Hydrodynamics," Brit. Ass. Rep., 1882, pp. 52...; Winkelmann, 

 Handbuch der Physik, Breslau, 1891..., t. i., p. 435. 



