RECENT ADVANCES IN SCIENCE 5 



M. Bauer (ibid. 90-103) gives his researches on the theory 

 of alegbraic corpora which appeared also in Math. Ann. (77, 

 353-61 ; Rev. sent. 25 [1], 26). 



G. H. Hardy and S. Ramanujan (Compt. Rend. 191 7, 164, 

 35-8) give an asymptotic formula for the number of partitions 

 of n. 



H. H. Mitchell (Amer. Journ. Math. 191 7, 39, 425-9) obtains 

 certain limits for the sum of all the positive integers less than 

 a given prime m, the indices of which, with respect to a primi- 

 tive root of m, have the same residue, mod 2w, where 2« is a 

 divisor of ra— 1. Perhaps the most interesting results concern- 

 ing the sums of the type considered were established by Dirich- 

 let in connection with his investigation of the class number 

 of quadratic forms, and Mitchell's results are obtained by use 

 of a certain type of Dirichlet's series. 



S. Composto (Giorn. di Mat. 19 16, 54, 290-93) has a note on 

 a determinant whose elements are functions of factorial powers 

 ('-' factorial Wronskian determinants "). 



C. E. Cullis (Bull. Calcutta Math. Soc. 191 6, 6, 33-48 ; 191 7, 

 8, 1-32) gives two parts of some researches on primitive 

 matrices and the primitive degrees of a matrix. 



J. B. Pomey (Nouv. Ann. de Math. 1916,16, 501-5), byanalogy 

 with the consideration (Cauchy, Kronecker) of imaginaries as 

 the residues of whole functions with respect to the modulus 

 1 + x 2 , calls by the name of " imaginary quantity " any whole 

 function of the undetermined x, if we consider it as " equiva- 

 lent " to the remainder after division by (x—Wi) (x—w 2 ) (x~w s ). 



G. A. Miller (Amer. Journ. Math. 191 7, 39, 404-6) proves 

 that it is possible to construct a non-abelian group which has 

 for its characteristic operators (those operators which corre- 

 spond to themselves in every possible automorphism of the 

 group) all the operators of an arbitrary abelian group. Miller 

 (Bull. Amer. Math. Soc. 191 8, 24, 203-6) exhibits the equivalence 

 of the theory of imprimitive substitution groups and the theory 

 of multiplication of another type of special matrices. 



Analysis. — O. Szasz (Math, es term. ert. Budapest, 191 5, 33, 

 654-83) has a paper on the convergence of continued fractions 

 of real elements. 



H. A. Kempner (Annals of Math. 191 7, 19, 127-36) investi- 

 gates a problem on " lattice-points " by very elementary 

 methods ; its complete solution is given by a pointwise dis- 



