139-140] MOVING SPHERE IN CYCLIC REGION. 217 



coordinates ^, ^', ..., viz. the virtual moment of this system is zero for any 

 infinitely small displacements of the solids, so long as ^, x', do not vary. 

 We may imagine, for example, that the impulses are communicated to the 

 membranes by some mechanism attached to the solids and reacting on these *. 

 Denoting these components by X, X', ..., and considering an arbitrary variation 

 of x, x', ... only, we easily find, by an adaptation of the method employed near 

 the end of Art. 134, that 



X'+ .................. (ii), 



whence the results (i) follow. 



The same thing may be proved otherwise as follows. From the equations 

 (2) and (1) of Art. 138, we find 



da> 



since v 2 o> = 0. The conditions by which o> is determined are that it is the 

 value of 3> when 



& = 0, ft = 0, ..., x = l, x' = 0, ..., .................. (iv), 



i.e. o) is the velocity-potential of a motion in which the boundaries, and 

 therefore also the barriers, are fixed, whilst 



-//*-* -//*'-*- ............... w- 



Hence the right-hand side of (iii) reduces to p*, as was to be proved. 



140. A simple application of the equations (21) of Art. 137 

 is to the case of a sphere moving through a liquid which circulates 

 irrotationally through apertures in a fixed solid. 



If the radius (a, say) of the sphere be small compared with its least 

 distance from the fixed boundary, then C, the kinetic energy of the system 

 when the motion of the fluid is acyclic, is given by Art. 91, viz. 



2&=m(x 2 +f+?) .............................. (i), 



where m now denotes the mass of the sphere together with half that of the 

 fluid displaced by it, and x, y, z are the Cartesian coordinates of the centre. 

 And by the investigation of Art. 99, or more simply by a direct calculation, 

 we have, for the energy of the cyclic motion by itself, 



2^= const. -27rpa 3 ^ 2 + v 2 + ^ 2 ) ..................... (ii). 



Again the coefficients a lt a 2 , a 3 of Art. 137 (18) denote the fluxes across 

 the first barrier, when the sphere moves with unit velocity parallel to #, y, z, 

 respectively. If we denote by O the flux across this barrier due to a unit 

 simple-source at (,r, y, 2), then remembering the equivalence of a moving 

 sphere to a double-source (Art. 91), we have 



fll = |a 3 dQ/dxy a 2 = %a s dQ/dy, a 3 = %a 3 dQ/dz ............... (iii), 



* Burton, Phil. Mag., May, 1893. 



