  The mechanics and statistical mechanics of a suspension of active particles
are determined by the traction (force per unit area) on their surfaces. Here we
present an exact solution of the direct boundary integral equation for the
traction on a spherical active particle in an imposed slow viscous flow. Both
single- and double-layer integral operators can be simultaneously diagonalised
in a basis of irreducible tensorial spherical harmonics and the solution, thus,
can be presented as an infinite number of linear relations between the harmonic
coefficients of the traction and the velocity at the boundary of the particle.
These generalise Stokes laws for the force and torque. Using these relations we
obtain simple expressions for physically relevant quantities such as the
symmetric-irreducible dipole acting on, or the power dissipated by, an active
particle in an arbitrary imposed flow. We further present an explicit
expression for the variance of the Brownian contributions to the traction on an
active colloid in a thermally fluctuating fluid.
