In the present study two sets of unbalanced two-way cross-classification data with and without empty cell(s) were used to evaluate empirically the various sums of squares in the analysis of variance table. Searle(1977) and Searle et.al.(1981) developed a method of computing R($\alpha

\mu, \beta$) and R($\beta

\mu, \alpha$) by the use of partitioned matrix of X'X for the model of no interaction, interchanging the columns of X in order of

and accordingly the elements in b. An alternative way of computing R($\alpha

\mu, \beta$), R($\beta

\mu, \alpha$) and R($\gamma

\mu, \alpha, \beta$) without interchanging the columns of X has been found by means of,

derived, using

. It is true that $R(\alpha

\mu,\beta,\gamma)\Sigma = SSA_W and R(\beta

\mu,\alpha,\gamma)\Sigma = SSB_W$ where

and means analysis and $R(\gamma

\mu,\alpha,\beta) = R(\gamma

\mu,\alpha,\beta)\Sigma$ for the data without empty cell, but not for the data with empty cell(s). It is also noticed that for the datd with empty cells under W - restrictions $R(\alpha

\mu,\beta,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\alpha

\mu) and R(\beta

\mu,\alpha,\gamma)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W = R(\beta

\mu) but R(\gamma

\mu,\alpha,\beta)_W = R(\mu,\alpha,\beta,\gamma)_W - R(\mu,\alpha,\beta,\gamma)_W \neq R(\gamma

\mu,\alpha,\beta)$. The hypotheses

commonly tested were examined in the relation with the corresponding sums of squares for $R(\alpha

\mu), R(\beta

\mu), R(\alpha

\mu,\beta), R(\beta

\mu,\alpha), R(\alpha

\mu,\beta,\gamma), R(\beta

\mu,\alpha,\gamma), and R(\gamma

\mu,\alpha,\beta)$ under the restrictions.