A method has been devised for solving the hydrodynamic problem involving swarms of particles moving in an incompressible fluid, assuming that the inertia terms in the equation of motion can be neglected. It is used to calculate the effective viscosity of a supension of spherical particles whose volume concentrations are less than. 10 %, assuming a statistical distribution of particles, under circumstances similar to those in a rotating viscometer. The viscosity is determined from the rate of shear at the walls. A special form of the model gives values of the effective viscosity up to concentrations as high as 25 % which are in good agreement with experimental values. The mathematical treatment involves the calculation of the potential due to point charges amongst a collection of earthed conducting spheres, coinciding with the particles. The simpler approximations allowing only for the induced charge on these spheres and the dipole moments about their centres are sufficiently accurate for concentrations of less than 10%, but a more complicated estimate is required for greater concentrations, which includes the effect of quadrupole and other multipole moments. Such calculations are elaborate but they are completed by means of expansions in solutions of Laplace’s equation V
2
0 = 0 which are derivatives of the elementary solution (1/r), instead of the more usual expansion in Legendre polynomials. A suitable algebra for these solutions is evolved which enables us to avoid problems which arise in changing the origin and axes of the Legendre polynomials. Using this algebra solutions of the equation V
2
0 =
k
2
0 are manipulated almost as easily as those of Laplace’s equation. In the Anal results for the effective viscosity, the effect of the multipoles is found to be small.
Derivatives of addition theorems for Legendre functions
