The factors associated with σ²_d and σ²_s

I rechecked a couple of sources and confirmed that my notes are correct. The expected mean square for the among dam component of variance is

σ²_w + kσ²_d

and the among sire component of variance is

σ²_w + kσ²_d + dkσ²_s

where k is the number of progeny per dam and d is the number of dams. The reason for the factor kσ²_d is that each of the k progeny of a single dam contribute to the among dam component of variability. Similarly, each of the dk progeny of a single sire contribute to the among sire component of variability.

I know it's a little strange, but that's the way it works out.