A zonal harmonic is a spherical harmonicof the form , i.e., one which reduces to a Legendre
polynomial (Whittaker and Watson 1990, p. 302). These harmonics are termed
"zonal" since the curves on a unit sphere
(with center at the origin) on which vanishes are parallels of latitude which divide the surface into zones
(Whittaker and Watson 1990, p. 392).

Resolving
into factors linear in ,
multiplied by
when
is odd, then replacing by allows the zonal harmonic to be expressed as a product of factors linear
in ,
, and , with the product multiplied by when is odd (Whittaker and Watson
1990, p. 1990).