Algebraic processes for determining the values of rational functions derived from power series have been compared in terms of operation counts by the reviewer [Math. Comput. 44, 147-186 (1960; Zbl 0173.18803)]. It is found that for the determination of one value in isolation, use of linear algebraic equations is most economical, that with regard to diagonal sequences of corresponding and associated continued fraction convergents, a recursive orthogonalisation process is the most suitable and that for a two dimensional array of values use of the epsilon algorithm is most appropriate. The theory of rational approximation extended by the reviewer to power series with coefficients over a ring; in particular an extended version of the orthogonalisation process has been obtained. A theory of continued fractions derived from power series with vector valued coefficients has also been introduced by the reviewer; in particular a generalised inverse over such power series has been defined [Compositio Math. 23, 453-460 (1971; Zbl 0239.15003)] as has an algorithm for the derivation of continued fraction coefficients from power series counterparts [viz. the conference proceedings Rocky Mt. J. Math. 4, 297-323 (1974; Zbl 0302.65005)]. None of quoted papers is mentioned in the paper under review, in which a vector version of the orthogonalisation process is derived. There are academic back scratching clubs whose members blithely attribute classical and not so classical results from the literature either to each other or to themselves. Had the author been a little more scrupulous in revealing his sources, the exposition would certainly have been clearer and the paper itself possibly of greater interest. Although orthogonalisation is effective in terms of counting algebraic operations, the resulting numerical process is highly unstable: use of a stable process such as the epsilon algorithm is to be preferred. It has been suggested that the vector epsilon algorithm is inherently an unstable process; the suggestion is repeated in this paper. The vector epsilon algorithm is in fact a high precision instrument which when used clumsily can produce unfortunate results, but when applied in appropriate circumstances is as effective and stable as in the scalar case. The interested research worker can confirm this assertion by applying the algorithm to the successive partial sums of the series whose vector terms \( {\mathbf t}(i)\) \((i \geq 0) \) have components \( t(i|k)=(-)^{i}v(i+k) \) for \( 0 \leq k \leq n \), where \( v(m)\) \((m \geq 0) \) is a moment sequence with members such as \( 1/(m+1) \) or \( 4/(2m+1) \) for example. The \( {\mathbf t}(m) \) are vector valued moments, accompanied by sign oscillation. Successive components of the vector series converge to differing limits that are simply related. The computations are completely stable. The algorithm induces marked convergence acceleration. The components of members of step sequences taken from the epsilon vector array alternately lie consistently above or consistently below corresponding components of the limit vector: the algorithm provides inclusion intervals for the limit. There is an extension to vector valued continued fractions of the theory concerning rational functions derived from power series in the scalar case, given by the reviewer in [Acta Math. Sci. Hung. 25, 291-298 (1974; Zbl 0323.30043)].