RECENT ADVANCES IN SCIENCE 193 



this general theorem. H. B. C. Darling (ibid. 217-8) also 

 gives proofs of these two particular cases. G. H. Hardy (ibid. 

 21 1-6) communicates, with an introductory note, simple proofs 

 of certain identities in combinatory analysis by L. J. Rogers 

 and S. Ramanujan. Rogers first discovered the identities in 

 a paper published in 1894, and Ramanujan rediscovered them 

 in 191 3 ; since then three proofs have been published, but the 

 present two proofs — which are in principle the same — are much 

 simpler than any published hitherto. 



Cf. also J. G. van der Corput (48), A. L. Bartelds and F. 

 Schuh (49), O. Szasz (13), G. Humbert (33, 34, 35), W. Janichen 

 (13), N. G. W. H. Beeger (45), V. Brun (11), S. C. van Veer 

 (50), G. H. Hardy (9), P. A. MacMahon (40), E. Maillet (37), 

 E. Hecke (18 [two papers], 29), E. Landau (15, 16 [two papers], 

 18 [two papers], 24, 27), G. Polya (13, 20, 26). 



Algebra. — G. A. Miller (Journ. Indian Math. Soc, 1919, 11. 

 57-9) gives a new interpretation of the ordinary complex 

 numbers a + hi as the translations in a plane defined by 

 x' = x + a and y' = y + b. In his words this is a new " proof 

 of the legitimacy " of complex numbers which " is based largely 

 on the group concept, but can clearly be given without the 

 explicit use of this concept." G. W. Smith (Amer. Journ. Math., 

 1 919, 41, 143-64) discusses nilpotent algebras generated by 

 two units i and j such that t 2 is not an independent unit. 



G. A. Miller (Quart. Journ. Math., 191 8, 48, 147-50) gives 

 some theorems relating to substitution-groups on the terms of 

 symmetric polynomials. 



Sir Thomas Muir (Proc. Roy. Soc. Edinburgh, 191 9, 39, 35-40) 

 gives a note on the determinant of the primary minors of 

 a special set of (n — 1) — by — n arrays. W. H. Metzler (ibid. 

 41-7) exhibits the rational and real factors of certain forms of 

 circulants. 



H. W. Turnbull (Proc. Lond. Math. Soc, 191 9, 18, 69-94) 

 shows how Gordan's system of invariants for two quaternary 

 quadratics can be very much simplified ; in fact, the system 

 is reduced to 125 forms instead of Gordan's 580 forms. Turn- 

 bull (Proc. Cambridge Phil. Soc, 1919, 19, 196-206), in connec- 

 tion with this reduction, gives geometrical interpretations to 

 most of the members of this system (which is here stated to 

 number " 123 at most "). 



Cf. also D. R. Curtiss (10), O. E. Glenn (8), I. Schur (26-7, 



