A Newton's endoreversible engine is defined when the heat flux from it to the cold bath J(c) obeys Newton's law, and the heat flux from the heat bath into the engine J(h) is arbitrary. For such a system a functional relationship between the power output function and the ecological function is found. Use is made of this relation to establish bi-univocal relations between two operational modes: at maximum power output (MPO) or at maximum ecological function (MEF). Bi-univocal relations are found between the engine's temperatures of the endoreversible engine and also between efficiencies at both operational modes. It is also shown that under very general conditions the upper engine's temperature and the efficiency in the MEF mode are greater than the upper engine's temperature and efficiency in the MPO mode, respectively. (C) 2010 Elsevier B.V. All rights reserved.