In this paper, some new linear relations for MacRobert's$E$-Functions are established. They are formulae (1), (10), (15), (20), (22) and (28) below. For the definitions and properties of these functions, the reader is referred to MacRobert, “Functions of a Complex Variable” (3rd ed., London 1946), p. 348. This work will be denoted by the letters C.V. Also some expansions of Bessel Functions are deduced.