In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.

1. A well known theorem in classical dynamics tells us that if Q, P, and are written as functions of q, p and t by means of the equations

where W (q, Q, t) is an arbitrary function, and if two functions F (Q, P, t) and f (q, p, t) are related by

then q and p are conjugate with respect to f(q, p, t) if Q and P are conjugate with respect to F (Q, P, t).

The chain of circle theorems now associated with the name of de Longchamps may be stated as follows.

(i) Any two lines L (A), L (B) meet in a point (AB).

(ii) Any three lines L(A), L (B), L(C) determine three points (AB) (BC) (CA) which lie on a circle S (ABC) with centre (ABC),

(iii) Any four lines L(A), L(B), L(C), L(D) determine four circles S(ABC) S(ABD) S(ACD) S(BCD) which meet in a point P (ABCD) and four centres (ABC) (ABD) (ACD) (BCD) which lie on a circle S(ABCD) with centre (ABCD).

(iv) Any five lines L(A), L(B), L(C), L(D), L(E) determine five points like P (ABCD), which lie on a circle SM (ABCDE) (Miquel's Theorem), five circles like S (ABCD), which meet in a point P (ABCDE), and their five centres like (ABCD), which lie on a circle S (ABCDE) with centre (ABCDE).

(v) Any six lines L (A), … . , L (F) determine six circles like SM (ABCDE) which meet in a point PM (ABCDEF), six circles like S (ABCDE), which meet in a point P (ABCDEF) and six centres like (ABCDE) which lie on a circle S (ABCDEF) with centre (ABCDEF).

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.

The six quaternary quartic forms

were first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

There is a group of Tauberian theorems of which the simplest isone due to K. Ananda Rau [Theorem 2 of the paper numbered 1 inthe list of references at the end of the note]. More complicatedtheorems of the same group are discussed in a paper by S.Minakshisundaram and myself to be published by the LondonMathematical Society [4].

The purpose of this brief note is to draw attention to a type of inverse factorial series which, so far as the writer can judge, has not been intensively studied. The central difference formulae of interpolation and the corresponding infinite series in central factorial polynomials have in the past received much attention, and so also has the ordinary inverse factorial series

which from the time of Stirling (Methodus Differentialis, 1730) has been known to be of value in transforming slowly convergent series into series of more rapid convergence.