RECENT ADVANCES IN SCIENCE 



PURE MATHEMATICS. By Dorothy Wrinch, University College, 

 London. 



G. W. Smith, American Journal of Math., xli. (1919), pp. 143- 

 164, discusses nilpotent algebras generated by two units : the 

 paper is based on the researches of Shaw, Trans. American 

 Math. Soc, iv (1903), pp. 251-287, 405-422. 



H. Holden, Messenger, xlviii (1919), pp. 77-87, discusses 

 a diophantine problem in three unknowns. 



G. Humbert, Comptes Rendus, 168 (1919), pp. 917-923, 969- 

 975, investigates the measure of classes of positive ternary 

 quadratic forms with a given determinant. His results are 

 extensions of the classical formulae due to Eisenstein and 

 H. J. S. Smith ; the same author studies the measure of 

 Hermitian forms, ibid., pp. 1240- 1246, and Hermitian forms in 

 a quadratic imaginary field, ibid., 169 (1919), pp. 360-365, 407- 

 414. 



J. E. McAtee, American Journal of Math., xli (1919), pp. 

 225-272, discusses modular invariants of a quadratic form for 

 a prime power modulus ; the first part of the paper is an 

 extension of L. E. Dickson's researches, Trans. American Math. 

 Soc, x (1909), P- 123, when the modulus is prime ; the second 

 part is occupied with the polynomial modular invariants of a 

 binary form. 



O. E. Glenn, Trans. Amer. Math. Soc, xx (1919), pp. 154- 

 168, investigates certain modular concomitant scales of the 

 binary quadratic ; the same author, ibid., pp. 203-212, investi- 

 gates a new treatment of theorems of finiteness of algebraic 

 invariant systems. 



A. Lodge, Messenger, xlviii (1918), pp. 101-107, discusses 

 the connexion between the cross ratio of the roots of a quartic 

 and the roots of its reducing cubic. 



G. A. Miller, Trans. Amer. Math. Soc, xx (19 19), pp. 260- 

 270, investigates the properties of groups which possess a small 

 number of conjugate operators ; the same author, Bull. Amer. 

 Math. Soc, xxv (191 9), pp. 408-413, investigates groups con- 

 taining operators of order two. 



G. Giraud, Comptes Rendus, 169 (191 9), pp. 1 31-134, in- 

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