MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
4G9 
in the case of u x , u 2 , consists of two parts, depending on the integration over the two 
spheres. As in the case of L this coefficient 1/ is 
L'v^ = — inv x X j p/+ z/ (jpj | —girv a s| } 
dashed letters referring to the motion of B, /x, v referring to images within A and B 
respectively, and a' cr denoting distances from the centres of A and B. This may be 
reduced as in the former case to 
L v x v 2 ^^ 477 v x ^ x (^jx ) 47 tv 2 £ x {v^ 
the fx' being the images in A of B’s motion perpendicular to A B, and v the images in 
B of As motion perpendicular to A B. 
L/tY^o— — AnV l V () 'Z x ) — 477 v. 2 [x () X 
where v 0 , fx 0 are the original doublets at the centres of B, A. i.e. 
_ V i i\ _ a?v x 
whence 
l '=1ma(9+*m iX (3 
in which last the ratios —, — do not contain v x , v 2 . 
v o H'o 
In the case of general motion of two spheres, each will have three components of 
velocity, u x , v x , w x ; u 2 , v 2 , w 2 ; and, in general, the expression for the kinetic energy 
will contain 21 terms. In the case in question we can easily see that the coefficients 
of 12 of these vanish. For consider the term in u x , iv 2 —suppose v x , w x , u 2 , v 2 all zero; 
the energy, from the symmetry of the motion, must clearly be unaltered if we reverse 
the direction of v\ 2 . And this can only happen if the coefficient of u^w 2 — 0. In this 
way we find the terms all vanish except those in u x , v x , iv x , u 2 , v 2 , w 2 , u x u 2 , v x v 2 , 
w x w 2 . Also from symmetry the coefficients of v x , v 2 , v x v 2 , are equal respectively to 
the coefficients of w x , w 2 , w x w 2 . 
In what has gone before we have expressed the coefficient of u x , u 2 , u x u 2 in terms 
of quantities determined by the radii and distance of the spheres, and have shown 
how the coefficients of the other terms depend on the images of the motion, whereby 
we can without much difficulty approximate to their values when the distance of 
the spheres is large compared with their radii—or the distance between their 
surfaces is large compared with the radius of one of the spheres. We pass on to 
MDCCCLXNX, 3 P 
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