Our model for an observation with the *i*th treatment in the *j*th block is

\(y_{ij}=\mu+\theta_i+u_j+e_{ij}\)

where the \(\theta_i\) are the treatment effects and the \(u_j\) are the block effects.

So let's set up the expression for the covariance between two observations from the same block, say block *j*, but different treatments, say *1* and *2*, as:

\(\text{cov}(y_{1j}, y_{2j})=\text{cov}(\mu+\theta_1+u_j+e_{1j}, \mu+\theta_2+u_j+e_{2j})\)

And now you can use the rules about covariances of linear combinations to simplify this expression.