1879.] 



the Motion of Tivo Spheres in a Fluid. 



163 



expressed in terms of the magnitudes of the doublets also supposed 

 known. The kinetic energy of the fluid motion is of the form 



2T = A^ 2 + Bx^! 2 + w?) + A 2 ^ 3 2 + B 3 (^ 3 2 + w 2 2 ) — 23W* 2 + 2M(v 1 v i + wtw t ) r 



where u h u 2 denote velocities along the line of centres, and v lt Wi . . '. . 

 perpendicular to it, and it is shown that 



A^m^l + 32^}, 



where m x is the mass of fluid displaced by the sphere, and ^™fi denotes 

 the sum of all the doublet- masses within (1) due to a unit motion of 

 (1) along the line of centres. A similar formula holds for B b the 

 in this case denoting the doublet-masses within (1) due to a unit 

 motion of (1) perpendicular to the line of centres. Also that 



L=|m 2 2^ O') + %n^(v), 



where denotes summation for all the images inside (2) due to 



a unit motion of (1) along the line of centres, and 2j°(V) summation 

 of all the images in (1) due to a corresponding motion of (2). An 

 analogous expression holds for M. A 1? A 2 , L, are completely deter- 

 mined in terms of the radii of the spheres and their distances, and an 

 approximation is made to the values of B 1? B 3 , and M. In certain 

 cases the values for the coefficients take up simple forms ; for instance, 

 at the moment when one sphere is concentric with the bounding one 



A 1 =-m 1 ^^j^ , a result already obtained by Stokes. 



The motion of the spheres along their line of centres is considered, 

 and it is shown that whatever be their velocities, the effect of the fluid 

 motion is such that relatively they appear to repel one another. More 

 particularly too is considered the case of motion of a single sphere in 

 an infinite fluid bounded by a plane. Amongst other things, it is 

 shown that the ratio of the limiting velocity to the velocity when in 

 contact with the plane = /y/ ^ ^^^061707 | ^ ^ ^g^g ^ e d ens ity 



of the sphere. 



Finally also is considered the effect of the fluid on vibratory motions 

 and the mutual influence of two pendulums when they oscillate in their 

 line of centres ; data are given in the paper for the determination of 

 the effect when they move in any manner. More particularly the effect 

 on a sphere is considered when another has impressed on it a small 

 harmonic vibration in the line of centres, and the force required to keep 

 it at rest. In the first case the mean effect is an acceleration towards 

 the vibrating sphere (A x ) 



