A form of integration of tensors will be introduced here, which will preserve the character of a tensor when so integrated.

We shall show in this part the relation of generalised C.F.'s to ordinary C.F.'s, in the main confining our attention to Stieltjes type fractions. Moreover we shall bring out the part played by Parseval's theorem in our development of the subject, and a property of extremal solutions of the Stieltjes moment problem given by M. Riesz.

We develop here the recurrence relations for the generalised C.F.'s introduced in Part 3 (Shenton 1956). In the main the discussion will be limited to second order C.F.'s, but results for higher orders will be given when these are not complicated.