In an earlier paper [1] the author presented a generalization of the second degree bipolykays of Hooke [2], defined for arbitrary balanced population structures, and showed the equivalence of these generalized polykays and the $\Sigma$ functions defined by Zyskind [7]. In this paper is presented a more general formalization of generalized symmetric means and polykays of arbitrary degree and some sampling properties of these. Utilizing the fact that the second degree generalized polykays are equivalent to the $\Sigma$'s, which are defined in terms of components of variation, an application to obtaining the variances of estimates components of variation is also presented.