  Any eigenfunction of the laplacian on the sphere is given in terms of a
unique set of directions: these are Maxwell's multipoles, their existence and
uniqueness being known as Sylvester's theorem. Here, the theorem is proved by
realising the multipoles are pairs of opposite vectors in Majorana's sphere
representation of quantum spins. The proof involves the physicist's standard
tools of quantum angular momentum algebra, integral kernels, and gaussian
integration. Various other proofs are compared, including an alternative using
the calculus of spacetime spinors.
