In this paper we define the concept of orthogonality between two factors ”through another factor”. Exploiting this property we have been able to obtain orthogonal main effect plans (OMEP) on non-orthogonal blocks requiring considerably smaller number of blocks than the existing methods. We have also constructed saturated partially orthogonal main effect plans (MEPs) for (i) an n.2 experiment and (ii) an n.2.3 experiment both on 4n runs. Here n is an integer ≥ 3, n 6= 4. As particular cases, we have been able to accomodate four six-level factors on 8 blocks of size 4 each using the first method and on 24 runs using the second. AMS Subject Classification : 62k10.