  The concept of orthogonality through the block factor (OTB), defined in
Bagchi (2010), is extended here to orthogonality through a set (say S) of other
factors. We discuss the impact of such an orthogonality on the precision of the
estimates as well as on the inference procedure. Concentrating on the case when
$S$ is of size two, we construct a series of plans in each of which every pair
of other factors is orthogonal through a given pair of factors.
  Next we concentrate on plans through the block factors (POTB). We construct
POTBs for symmetrical experiments with two and three-level factors. The plans
for two factors are E-optimal, while those for three-level factors are
universally optimal.
  Finally, we construct POTBs for $s^t(s+1)$ experiments, where $s \equiv 3
\pmod 4$ is a prime power. The plan is universally optimal.
