In this paper for equimorphic [2] functions two theorems are given about Gavrilov`s P-sequence [3] and Collingwood`s maximality theorem ([1], p.108) along lines that have arbitrary tangential order with the unit circle. Afterwards Meier`s theorem is strengthened ([1],p.204) for equimorphic functions, as well as Dolzhenko`s theorem ([1],p.249) along arbitrary tangential directions is generalized for arbitrary functions.