20G MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



wall. The reason of this is easily seen by reducing the problem to 

 one of steady motion. The fluid velocity will evidently be greater, 

 and the pressure, therefore, will be less, on the side of the sphere 

 next the wall than on the further side ; see Art. 24. 



The above investigation will also apply to the case of two 

 spheres projected in an unlimited mass of fluid, in such a way 

 that the plane y = is a plane of symmetry as regards the motion. 



136. Let us next take the case of two spheres moving in the 

 line of centres. 



The kinematical part of this problem has been treated in Art. 97. If we 

 now denote by #, y the distances of the centres A, B from some fixed origin 

 in the line joining them, we have 



where the coefficients Z, J/, N are functions of c, =y - x. Hence the equations 



of motion are 



/7 \f ,7 AT 



dt 

 d_ 

 dt ( " '"*' *\dc~ " dc 



where JT, Y are the forces acting on the spheres along the line of centres. If 

 the radii a, b are both small compared with c, we have, by Art. 97 (xv), 

 keeping only the most important terms, 



,-.3 A3 



(iii) 



approximately, where m, m' are the masses of the two spheres. Hence to 

 this order of approximation 



dL dM a?V dN 



~j~ = "j ~~T~ = ~ Virp T~ i T~ == " 



dc dc c 4 ' do 



If each sphere be constrained to move with constant velocity, the force 

 which must be applied to A to maintain its motion is 



dM . dM . 



This tends towards B, and depends only on the velocity of B. The spheres 

 therefore appear to repel one another ; and it is to be noticed that the apparent 

 forces are not equal and opposite unless #= +y. 



Again, if each sphere make small periodic oscillations about a mean 

 position, the period being the same for each, the mean values of the first terms 

 in (ii) will be zero, and the spheres therefore will appear to act on one another 



with forces equal to 



a 3 ft 3 

 GTTP ^ [ay] .............................. (v), 



