  We review Maxwell's multipole vectors, and elucidate some of their
mathematical properties, with emphasis on the application of this tool to the
cosmic microwave background (CMB). In particular, for a completely random
function on the sphere (corresponding to the statistically isotropic Gaussian
model of the CMB), we derive the full probability density function of the
multipole vectors. This function is used to analyze the internal configurations
of the third-year Wilkinson Microwave Anisotropy Probe quadrupole and octopole,
and we show the observations are consistent with the Gaussian prediction. A
particular aspect is the planarity of the octopole, which we find not to be
anomalous.
