§ 1. Introductory. In § 3 a generalisation of the formula [MacRobert, Phil. Mag., Ser. 7, XXXI, p. 258]
where αp+1 = ½m + ½n, αp+2 = ½m - ½n, R(m ± n) > 0, and x is real and positive, will be established. In the course of the proof Hardy's formula [Mess, of Maths., LVI, (1927), p. 190],
where R(b)>0, will be required. This was originally proved by an application of Mellin's Inversion Formula. An alternative proof is given in § 2, and some related formulae are deduced.