346 SCIENCE PROGRESS 



aspects of the same fundamental property, or that there are 

 two fundamental properties involved in such a manner that one 

 of the postulates, at least, deals with both properties. J. S. 

 Taylor {Bull. Amer. Math. Soc, 26, 1920) modifies the first of 

 Sheffer's five postulates, and proves the complete independence 

 of the five postulates in their emended form. As Sheffer stated 

 them the negative of the first implies the third, fourth and fifth 

 postulates of the set. 



In the Proc. Roy. Soc. Edin., 40, 1920, F, L. Hitchcock, in 

 a paper entitled " An Identical Relation connecting Seven 

 Vectors," develops an identical equation satisfied by any seven 

 vectors, and also satisfied by an arbitrary quadratic function, 

 and explains methods by which new identities can be derived 

 from any vector identity. 



A. D. Pitcher (Bm//. Amer. Math. Soc, 26, 1920) discusses how 

 far the property of coherence (introduced and defined in a paper 

 by A. D. Pitcher and E. W. Chittenden, ibid., 19, 191 3) belongs 

 to the systems introduced by E. H. Moore {ibid.) in 1910. 



W. E. Milne {Bull. Amer. Math. Soc, 26, 1920) approaches the 

 study of infinite systems of function from an elementary point 

 of view, and derives results of considerable generality. The 

 discussion is limited to real functions of a single real variable, 

 but the methods used can be extended to more general 

 systems. 



Norbert Wiener {Ann. Math., 21, 1920), develops the neces- 

 sary and sufficient condition that a bilinear operation in two 

 variables should generate by iteration all rational operations 

 with rational coefficients. This is a particular case of the 

 general problem of determining what operations any given 

 bilinear operation will generate by iteration. In the Bull. Amer. 

 Math. Soc, 27, 1920, he develops methods of attack which yield, 

 in particular, an important necessary condition that each of 

 two operations generate the other by iteration. The complete 

 solution of the problem is still to be accomplished. 



J. Mercer, in a paper entitled " Symmetrisable Functions 

 and their Expansions in Terms of Biorthogonal Functions " 

 {Proc Roy. Soc, 93, 1920), announces certain results relative to 

 the expansion of a symmetrisable function k (s, /) in terms of 

 a complete biorthogonal system of fundamental functions 

 which belong to k (s, t) regarded as the kernel of a linear integral 

 equation. 



E. Borel, in a paper entitled, " Sur la classification des ensem- 

 bles de mesure nulle " {Bull. Math. Soc. France, 47, 191 9), studies 

 and classifies by means of the elementary properties of decimal 

 fractions certain classes of zero measure. 



