RECENT ADVANCES IN SCIENCE 539 



and (0) imply (K) when the path is the radius, and that (0) 

 and (o) in these statements can in fact be replaced by other 

 less restricted conditions of a different form. These are 

 perhaps the more interesting results in this difficult subject 

 which have been so far completely elucidated, but they are 

 not of course exhaustive. 



The authors begin the new work by an extension of one of 

 the theorems above — that (L) and (O) imply (K) when C is 

 the radius (0,1). They show in the first instance that this 

 radius can be replaced by a regular Stolz path. The main 

 problem undertaken, after this preliminary, concerns the 

 condition, which may be called (A), with which it is necessary 

 to replace (L) in order to obtain satisfactory results for regular 

 paths — that is to say, paths with a continuously turning tangent 

 at every point except x — i, and approaching x = i with a 

 definite direction. This condition (A) has been elucidated in 

 earlier work, and it was already known that (A) and (0) 

 imply (K) whenever C is a regular path in this sense. The 

 main problem may be described as the removal of this restriction 

 to regular paths, and the simultaneous replacement of (0) by 

 (0). This problem is solved, but the restrictions on the path 

 cannot be removed entirely when the condition (L) is in question, 

 though some important corollaries emerge which it is not 

 necessary to mention in detail here. The advance made is 

 very fundamental and should be a means of stimulating the 

 interest of other mathematicians, already growing, in problems 

 of this type. 



A side-issue of some considerable interest emerges in con- 

 nection with the simultaneous convergence of a Fourier series 

 and its conjugate or allied series, a matter which is of importance 

 in the applications of mathematics, and on which insufficient 

 attention is usually bestowed. Such conjugate series are 

 defined by the typical terms a n cos n6 + j3 n sin nO and a n 

 sin nd — /S„ cos nO, derived from the complex (a n + ifi n ) 

 exp. in6. 



The same number contains also a note by G. H. Hardy on 

 two problems in the analytic theory of numbers, and the 

 earlier part of a paper by E. L. Ince on some continued fractions 

 of a curious type associated with the hypergeometric equation. 



R. L. Moore, Amer. Journ. of Math. xli. (1919), pp. 299- 

 319, carries on an investigation of the Lee — Reemann — Helm- 

 holtz — Hilbert problem of the foundations of geometry. The 

 paper arises from a well-known remark of Poincar£, that 

 Hilbert's hypotheses are much more general than those of Lee, 

 who defined groups by analytic equations. But Poincare did 



